w. my»?! .2... .1 , 25.5.... l.. r 3 A i M11 . . .r. r. t. 1.... X}, z "-4.34: ‘ :1 .‘o~ 4113‘ .0 . t .1? '3. all 33. F s Ir$.i.1hfilzt..5. than“ . . .zErJ...‘ . S 2’! m i 1...... 3:. n . .lef z . a. .53.. ...~ 3m": W5 “Wihfinw In: 2... 1P. ., .5 a aweggz.‘ Egééwfifig 1*. 1. LIBRARIES MICHIGAN STATE UNIVERSITY EAST LANSING, MICH 48824-1048 This is to certify that the dissertation entitled THE INCLUSIVE JET CROSS SECTION IN PROTON- ANTIPROTON COLLISIONS AT A CENTER OF MASS ENERGY OF 1.96 TeV presented by GENE U. FLANAGAN has been accepted towards fulfillment of the requirements for the PhD degree in PHYSICS a @or Professor’s Signature L//" ,0 ),.-—— / - Date MSU is an Affinnative Action/Equal Opportunity Institution PLACE IN RETURN BOX to remove this checkout from your record. TO AVOID FINES return on or before date due. MAY BE RECALLED with earlier due date if requested. DATE DUE DATE DUE DATE DUE 2/05 cJCIRfiataDueJndd-p. 1 5 THE INCLUSIVE JET CROSS SECTION IN pp COLLISIONS AT \/5 = 1.96 TeV By Gene U. Flanagan A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Physics and Astronomy 2005 ABSTRACT THE INCLUSIVE JET CROSS SECTION IN pp COLLISIONS AT \/E = 1.96 TeV By Gene Flanagan The following work presents a preliminary measurement of the inclusive jet cross section for jet transverse momenta from 61 to 620 GeV in the rapidity range 0.1 < |Y| < 0.7. The result is based on 218 pb"1 of data collected by the CDF detector at the Fermi National Accelerator Lab. The data are consistent with NLO pQCD predictions based on the CTEQ6.1 parton distribution functions. ACKNOWLEDGEMENTS I would like to thank everybody who has been involved with the analysis for their help, advice and direct contributions. Many thanks to my supervisor, Joey Huston, for his insight, guidance, assis- tance and patience with me throughout the analysis. I would also like to thank Carl Bromberg, who assisted me in a number of ways during my time at Michigan State University. There are a number of people who not only contributed to the analysis but also to my continuing education: Anwar Bhatti, who took on the unenviable task of helping me learn and use C++ and for his guidance and assistance throughout the analy- sis; Frank Chlebana, Kenechi Hatakeyama and Giuseppe Latino for their assistance, advice and contributions. I would also like to thank the CDF collaboration in its entirety-it takes a lot of effort by many people to make any analysis possible. Finally I thank my family: my wife Amy, who has been incredibly supportive and has had to endure the analysis as much as anybody else—in some ways more as she doubled as my editor, my parents, Kevin and Elizabeth, and my brother, Jack, who have always been supportive and without whom I would not have come this far. iii Contents LIST OF TABLES .............................. viii LIST OF FIGURES (Images in this dissertation are presented in colour) x 1 QCD Theory 1 1.1 Introduction ................................ 1 1.2 The QCD Lagrangian ........................... 2 1.3 Green’s Functions & Observables .................... 5 1.3.1 Feynman Rules & Green’s thctions .............. 5 1.3.2 The S-Matrix and Cross Sections ................. 6 1.3.3 Divergences and Renormalisation Schemes ........... 8 1.3.4 Renormalisation Scale and Experimental Results ........ 11 1.4 Asymptotic Freedom ........................... 12 1.4.1 Forces in QCD .......................... 12 1.4.2 The Renormalization Group and the Effective Coupling . . . . 16 1.5 Infrared Safety .............................. 18 1.6 Jet Production .............................. 23 2 Jet Identification 29 2.1 Introduction ................................ 29 3 Jet Algorithms 34 3.1 Introduction ................................ 34 3.2 Theoretical Attributes of a Jet Algorithm ................ 34 3.3 Experimental Attributes of a Jet Algorithm .............. 35 3.4 JetClu ................................... 36 3.4.1 PreClustering ........................... 36 3.4.2 Clustering ............................. 37 3.4.3 Merging and Splitting ...................... 37 3.5 The Midpoint Jet Algorithm ....................... 38 3.5.1 Clustering ............................. 39 iv 3.5.2 Splitting and Merging ...................... 40 4 The Detector 41 4.1 Introduction ................................ 41 4.2 Experimental apparatus ......................... 41 4.2.1 The Accelerator Complex .................... 41 4.2.2 Protons .............................. 41 4.2.3 Antiprotons ........................ .. . . . 42 4.2.4 Collisions ............................. 43 4.3 The CDF Detector ............................ 43 4.3.1 Central Outer Tracker ...................... 45 4.3.2 Magnetic Field .......................... 46 4.3.3 Calorimetry ............................ 46 4.3.4 Resolution of the calorimeters .................. 47 4.3.5 Cherenkov Luminosity Counters (CLC) ............ 48 4.3.6 Segmentation ........................... 48 5 The CDF Trigger 49 5.1 Introduction ............ A .................... 49 5.2 The CDF Trigger Architecture ...................... 49 5.2.1 The Level 1 Trigger ........................ 50 5.2.2 The Level 2 Trigger ........................ 51 5.2.3 The Level 3 Trigger ........................ 52 5.3 Trigger Efficiency ............................. 52 5.4 Prescales .................................. 56 6 Multiple Interactions & Underlying Event 59 6.1 Introduction ................................ 59 6.2 Data Set .................................. 60 6.3 Method .................................. 60 6.3.1 Minimum Bias Momentum in a Random Cone ......... 60 6.3.2 Summing of Tower Momentum in a Cone ............ 61 6.3.3 Effect of Single Tower Threshold ................ 62 6.4 Multiple Interaction and Underlying Event Subtraction ........ 62 6.5 Results ................................... 63 6.5.1 Corrections Versus Instantaneous Luminosity ......... 75 6.5.2 Instantaneous Luminosity & Number of Quality 12 Vertices in Jet Samples ............................ 6.6 90" Transverse Momentum (Pgo) in Jet Events ............ 6.7 Conclusion ................................. 7 Jet Energy Resolution 7.1 Introduction ................................ 7.2 Method .................................. 7.3 Results ................................... 7.4 Conclusion ................................. 8 The Raw Inclusive Jet Cross Section 8.1 Introduction ................................ 8.2 Data Sample ................................ 8.2.1 Run Selection ........................... 8.3 Event Selection .............................. 8.3.1 Z-Vertex Cut ........................... 8.4 Kinematics ................................ 8.5 Backgrounds ............................... 8.6 Raw Inclusive Jet Cross Section ..................... 9 Jet Corrections 9.1 9.2 9.3 9.4 Introduction ................................ Monte Carlo Simulation ......................... 9.2.1 Weighting the Monte Carlo ................... Average P71“ Correction ........................ 9.3.1 Calorimeter Level Thresholds at Hadron Level ......... Smearing Correction ........................... 9.4.1 P¥“’(Corr) Resolution and Bin Size .............. 9.4.2 Bin by Bin Smearing Correction ................. 9.4.3 Parton to Hadron Corrections .................. 10 Comparison of the Data to Pythia+CDFSIM 10.1 Introduction ................................ 10.2 Quantities of interest ........................... 10.3 Results ................................... vi 82 87 98 99 99 102 104 111 112 112 112 113 114 115 119 136 155 162 162 163 164 168 169 175 175 180 182 189 189 190 192 11 Comparison of the Data to NLO pQCD 215 11.1 Introduction ................................ 215 11.2 Correcting NLO pQCD for Underlying Event .............. 216 11.3 Results ................................... 217 11.4 Conclusion ................................. 227 12 Systematic Uncertainties 228 12.1 Introduction ................................ 228 12.2 Jet PT Scale Uncertainties ........................ 228 12.2.1 Calorimeter Response ....................... 229 12.2.2 Unfolding ............................. 229 12.2.3 Multiple Interaction Uncertainties ................ 230 12.2.4 Underlying Event ......................... 230 12.2.5 Integrated Luminosity and Z Vertex Uncertainty ........ 230 12.2.6 Results ............................... 231 12.3 Jet Energy Resolution Uncertainty ................... 237 12.4 Sensitivity to Input PDF ......................... 242 BIBLIOGRAPHY .............................. 248 vii List of Tables 1.1 4.1 5.1 5.2 6.1 7.1 7.2 7.3 7.4 7.5 8.1 8.2 8.3 8.4 8.5 8.6 9.1 9.2 10.1 11.1 Leading order jet production matrix elements squared (Z [M |2/g4) . 28 Segmentation of the calorimeters .................... 48 Trigger paths ............................... 52 Trigger efficiency and prescales for the jet triggers ........... 55 Underlying event and multiple interaction measurements for cone R = 0.7 ...................................... 63 KT“ and KTJ. jet and dijet PT selection cuts ............. 103 KT” for data and Pythia. ........................ 105 KT; for data and Pythia. ........................ 105 Jet energy resolution (UK M 5): data and Pythia ............. 106 Jet energy resolution (0R M 5): fractional difference between data and Pythia .................................... 106 Data sample before and after good run requirements. ......... 114 Number of events passing event selection cuts (ETotala ET & Z). . . . 115 Efficiency of Z vertex cut in the jet triggers. .............. 116 Raw inclusive jet cross section ....................... 156 Jet properties for jets with PT above 400 GeV .............. 158 Event properties for jets with PT above 400 GeV. ........... 160 Summary of Pythia bin correction factors ................ 187 Average P11“ corrected inclusive jet cross section. ........... 188 Ratio of measured hadron level inclusive jet cross section over Pythia hadron level cross section. ........................ 214 Inclusive jet cross section corrected to the hadron level and NLO pQCD prediction ................................. 223 viii 11.2 11.3 11.4 12.1 12.2 12.3 12.4 12.5 12.6 Hadron level Data over NLO pQCD including full statistical and sys- tematic errors ................................ 224 Hadron level Data over NLO pQCD including full statistical and sys- tematic errors ................................ 225 Hadron level Data (corrected for UE) over NLO pQCD including full statistical and systematic errors ...................... 226 Energy scale systematic uncertainty ................... 233 Hadronisation / unfolding model systematic uncertainty ........ 234 Multiple interaction systematic uncertainty .............. 235 Underlying Event systematic ...................... 236 Resolution systematic uncertainty ..................... 240 Total systematic uncertainty on the inclusive jet cross section . . . . 241 ix List of Figures 1.1 1.2 1.3 1.4 1.5 2.1 2.2 5.1 5.2 5.3 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 Feynman diagrams/ rules for the vertices ................. Lowest order QCD / QED diagrams (potential) and field theory correc- tions to the potential ............................ Feynman rules for a) quark propagator, b) ghost propagator and c) gluon propagator .............................. Diagrams that contribute to jet production ............... QCD subprocess contributions ...................... Lego Plot of CDF dijet event ...................... COT and calorimeter display of a CDF dijet event .......... Trigger efficiencies ............................ Relative prescales for Jet20, Jet50, Jet70 and Jet100 triggers ..... Inclusive jet PT spectrum for Jet20, Jet50, Jet70 and Jet100 after applying prescales ............................. Event properties of the minimum bias sample ............. Effect of tower threshold on the minimum bias sample. ........ PT in random cone R = 0.7 (tower threshold 100 MeV). ....... Linear fit to minimum bias momentum versus number of quality 12 vertices for tower thresholds of 50, 100 and 150 MeV ......... Linear fit to minimum bias momentum and energy versus number of quality 12 vertices (100 MeV tower threshold) ............. Instantaneous luminosity and number of quality 12 vertices versus luminosity for the minimum bias sample ................ Minimum bias momentum in a random cone as a function of quality 12 vertices in 8 bins of instantaneous luminosity (Pgowe’ > 100 MeV) Multiple interaction correction and underlying event as a function of the instantaneous luminosity ...................... Instantaneous luminosity of the jet trigger samples .......... 6.10 Number of vertices per event in the jet samples ............ 20 21 22 26 27 30 30 54 57 58 64 66 69 73 74 76 78 6.11 P1?~O(ave) and number of towers for 50—100 MeV tower thresholds . . . 6.12 P120(¢+90), P%0(¢—90), P%0(max), P:?~0(mz'n), P19»0(ave) and P199(dz' f f) for PTTW‘Br > 100 MeV .......................... 6.13 P%O(¢+90), P19~O(¢—90), P%0(ma:r), P%0(min), Pf?~0(ave) and P7910(dz' f f ) for PTTWC" > 50 MeV .......................... 6.14 P19~O(max), P%O(mz'n) and P%0(dz' f f ) for firm” > 100 MeV as func- tion of lead jet PT ............................ 6.15 P90(max), P%0(mz'n) and P199(dz' f f ) for B}: ”w" > 50 MeV as function 0 lead jet PT ............................... 7.1 PT balance in the transverse plane ................... 7.2 Data-Pythia Resolution difference versus (Pgijet) 7.3 KT”: data & Pythia ........................... 7.4 K11: data & Pythia ........................... 8.1 Z vertex of the four jet triggers before and after selection cuts. 8.2 Jet20 Kinematic variables: Inclusive PT , Y and d) .......... 8.3 Jet20 Kinematic variables: Etotal: ET and ET ............. 8.4 Jet50 Kinematic variables:Inclusive PT , Y and d) ........... 8.5 Jet50 Kinematic variableszEtoml, ET and ET ............. 8.6 Jet70 Kinematic variables:Inclusive PT , Y and <15 ........... 8.7 Jet70 Kinematic variables:Etotal, ET and ET ............. 8.8 Jet100 Kinematic variables:Inclusive PT , Y and 45 .......... 8.9 Jet100 Kinematic variableszEtotal, ET and ET ............. 8.10 Raw data distributions ET versus Leading jet for the jet20 trigger sample before and after selection cuts .................. 8.11 Raw data distributions EET versus lead jet for the jet20 trigger sample before and after selection cuts continued ................ 8.12 Raw data distributions ET versus Leading jet for the jet50 trigger sample before and after selection cuts .................. 8.13 Raw data distributions EET versus lead jet for the jet50 trigger sample before and after selection cuts continued ................ 8.14 Raw data distributions ET versus Leading jet for the jet70 trigger sample before and after selection cuts .................. xi 88 90 93 96 97 104 107 109 117 120 122 124 128 134 137 141 145 8.15 Raw data distributions BET versus lead jet for the jet70 trigger sample before and after selection cuts continued ................ 8.16 Raw data distributions ET versus Leading jet for the jet100 trigger sample before and after selection cuts .................. 8.17 Raw data distributions BET versus lead jet for the jethO trigger sam- ple before and after selection cuts continued .............. 8.18 ET before and after the ET cut for the jet20, jet50, jet70 and jet100 samples. ................................. 8.19 Uncorrected (calorimeter level) inclusive jet cross section as a function of PT .................................... 9.1 Number of events as a function of RT for Pythia after weighting. 9.2 Inclusive Pythia PT distribution for hadron level and calorimeter level jets after event weighting ........................ 9.3 Trigger efficiency cut in the hadron level distribution ......... 149 151 169 9.4 Pfiad/Pga’, YHad — Y0“’ and ¢H°d — (1)001 for matched jets (Pythia). 170 9.5 (Pfiadrm/PCGIO") versus P1001"; Pythia PT 2 10 GeV and the full Pythia samp e. .............................. 172 9.6 2D PT plot for matched jets in the full Pythia sample. ........ 173 9.7 (131}! ad) — P1?“ correlation used to derive the average PT,“ correction curve . ................................... 174 9.8 P7}! 0‘17”" distributions for a fixed values of P797010" .......... 176 9.9 mean(fb) from gaussian fits versus PT. ................ 177 9.10 0(fb) from gaussian fit versus PT ................... 178 9.11 a X qual(Corr) versus PT ....................... 179 9.12 Smearing correction (Pythia). ...................... 181 9.13 Fragmentation + underlying event correction (Pythia) ........ 184 9.14 Underlying event correction (Pythia) .................. 184 9.15 Fragmentation correction (Pythia) .................... 185 9.16 Fragmentation + underlying event correction (Herwig) ........ 186 9.17 Underlying event correction (Herwig) .................. 186 10.1 Z-Vertex distributions: data and Pythia ................ 192 10.2 ET distributions: data and Pythia .................... 194 10.3 ET distributions: data and Pythia .................... 196 xii 10.4 APT distirbutions: data and Pythia ................... 10.5 A45 distributions: data and Pythia ................... 10.6 KT" distributions: data and Pythia .................. 10.7 KT J. distributions: data and Pythia .................. 10.8 Electromagnetic fraction distributions: data and Pythia ........ 10.9 Rapidity (Y) distributions: data and Pythia .............. 10.10Azimuthal angle ((15) distributions: data and Pythia .......... 10.11Data corrected to hadron level versus hadron level pythia (CTEQ5L): log scale ................................... 10.12Data corrected to hadron level versus hadron level pythia (CTEQSL): linear scale. ................................ 11.1 Data corrected to hadron level versus NLO pQCD (no underlying event correction) ............................. 11.2 Data corrected to hadron level versus NLO pQCD (Linear)(no under- lying event correction) .......................... 11.3 Data corrected to hadron level versus NLO pQCD corrected for un- derlying event (Pythia). ......................... 11.4 Data corrected to hadron level before and after underlying event cor- rection versus NLO pQCD. . . . .. ................... 11.5 Data corrected to parton level versus N LO pQCD ........... 12.1 Systematic error contributions ........ I .............. 12.2 Difference in jet energy resolution between data and Pythia ...... 12.3 0(fb) versus P11! “drm ........................... 12.4 PT Pythia CTEQ51 Monte Carlo and re—weighted Pythia. ...... 12.5 Ratio of PT Pythia CTEQ51 and re—weighted Pythia. ......... 12.6 Calorimeter level inclusive jet distribution for Pythia CTEQ51 and Pythia re-weighted by CTEle ..................... 12.7 Hadron level inclusive jet distribution for Pythia CTEQ51 and Pythia re—weighted by CTEle ......................... 12.8 Corrected Pythia cross section over true Pythia cross section xiii 198 200 202 204 206 208 210 212 221 222 231 239 239 243 244 245 246 247 Chapter 1 QCD Theory 1.1 Introduction Our understanding of the physical universe is contained in the Standard Model, which describes the interactions between the known particles. These interactions are gov- erned by the four fundamental forces; 0 strong 0 weak 0 electromagnetic o gravitational. The fundamental constituents of the known particles and therefore of matter are quarks and leptons, both of which occur in three generations. Leptons are described by the following quantum numbers: spin, charge, baryon number, isospin and mass. Quarks too have these quantum numbers but additionally they carry a colour charge. In the case of quarks the baryon number is always i1 / 3. Quarks are the constituents of hadrons; hadrons are colourless particles that carry integer charge and baryon number. Hadrons are either mesons (quark-antiquark states, e. g. 7r+ (ud)) or baryons (3 quark configurations, e.g. p(uud) and n(’udd)). Baryons are fermions with a baryon number 1, whereas mesons are bosons with a baryon number 0. In addition to the valence quarks, hadrons also have a sea of quark-antiquark pairs and gluons, where the gluons are the force-mediating bosons of Quantum Chromo Dynamics (QCD). QCD is the accepted theory of strong interactions, which take place between quarks which make up the hadrons. 1.2 The QCD Lagrangian QCD is defined as a field theory by its Lagrange density: LéQfC‘; M[¢f($)a (1:) C(13); g: mf]= Linvar + Lgauge + Lghost, (1.1) which is a function of the quark field ’I/Jf, the gluon field A, the ghost field c, the coupling strength 9 and the fermion mass m f. The subscripts f label the distinct quark fields. Lima, is the classical density which is invariant under SU(Nc) gauge transformations. For QCD the number of colours describing quarks is 3 and so NC = 3. The classical density was originally written by Yang and Mills [1] in the form: Linvar = :Efli DUI] - mfill’f — iniA] s... 4 4 Nc _ = Z Z ,2 ‘l’fflalihl’éeDuJilAl “mf‘sfia‘sjiwfnd Ng-l 711: CE F,,,,G[A]F“"[A], (1.2) 4p,u=0 a=0 where f is the flavour, a and 3 are Dirac spinor labels, 2' and j are the colour labels, u and I/ are Lorentz indices and a labels the colour adjoint. In the above we have used the notation: Duel/4] E @1527 + i9Aua(Ta)ij, (1-3) pr,a[A] E aflAl/a _ ayAfla _ gCabCAflbAl/C’ (1.4) where Fm“, is the non—Abelian field strength tensor defined in terms of the gluon vector field A”, g is the strong coupling constant and the Cuba’s are real numbers which are the structure constants of the S U (NC) group. The Lie algebra is defined in terms of the commutation relations of the NC — 1, NC x NC matrices, (Ta),-j, [Ta, Tb] = iCabcTCa (1.5) where the Ta’s are matrices in the fundemental Nc-dimensional respresentation. Tak- ing the Ta’s to be hermitian in this representation makes QCD look very much like Quantum Electro Dynamics (QED). 1 D5} [A] is the covariant derivative in the Nc—dimensional representation of S U (NC), which acts on the spinor quark fields II), with colour indices 2' = 1...Nc. There are nf = 6 independent quark fields in the Standard Model, they are labelled by flavour: f(= u, d, c, s, t, b). The quark fields (2%) transform under local gauge transformations as 114,04- (37) = U jt' ($)¢f,a,i($)t (1-6) where (1.7) a Ng-l sz-(rc) = [exp {2' 2:1 fla(x)Ta} ji For every value of 2:, Uij is an element of the SU(NC) group, which is the local invariance that was built into the theory. The transformation of the gluon field is described in terms of the NC X NC matrix, Ap(:1:), N3—1 [A#(~T)iij 5 2:1 Apa($)(Ta)ij- (1-8) The gluon field is defined to transform according to Age) = U(e)A,,(e)U-1(e) + g[8pU(x)]U‘l(:r). (1.9) A mass gluon mass cannot be included in the Lagrangian, as a term of the form mzAflA" would violate the gauge invariance of the theory. The gauge particle, the gluon, must be massless. To facilitate the use of perturbation theory to make calculations of QCD quantities we need to fix the gauge. Without a gauge fixing term the gluon prOpagator has no inverse, rendering the use of perturbation theory impossible. There are different ways we can fix the gauge, all of them, however, break the gauge invariance of the theory. The breaking of the gauge invariance comes through the introduction of a parameter A. It does not matter which choice of gauge fixing term is chosen, as physical quantities (scattering matrix terms) do not depend on A, however, the intermediate steps of a calculation may look very different depending on the choice of gauge [2]. Gauge fixing can be achieved by either requiring a purely physical gauge or se- lecting the set of more general covariant gauges and introducing the associated ghost fields. The derivation of the form of the ghost field can be found with a path integral formulation [4, 6]. The ghost field will cancel the unphysical degrees of freedom that would otherwise propagate in a covariant gauge. The Lagrangians for the gauge fields are of the form: @3232“ = 5 Z (6,,Ag)2 and a=1 h ' z ANg—l 2 Lgaggzeca = '5 Z (na.Aa) , (1.10) a=l where n is a vector, /\ is a gauge parameter and A is the gluon field. The first of the above densities (Lfig’zem) defines a set of covariant gauges that can be added to the QCD Lagrangian (see equation 1.1). If this choice is made the ghost Lagrangian is of the form: Lghost = (BuEaXBWad - QCabdAg)cdv (1-11) where ca and Ea are the ghost and anti-ghost fields [9, 10]. The ghost fields anti- commute (even though they are scalar). Setting /\ = 1 in the set of covarient gauges gives the Feynman gauge. The Feynman gauge has a fairly simple gluon propagator (see figure 1.1). The second of the densities defines the set of physical gauges [11]. With this choice, the limit A —> oo eliminates the need for the introduction of a ghost density. The light-like n (n2 = 0) selected from this set is known as the light cone gauge. 1.3 Green’s Functions & Observables 1.3.1 Feynman Rules & Green’s Functions Choosing the generators T a to be hermitian, and taking the Fourier transform as- sociates everywhere 0,, -—> —iqp, where qp is the momentum flowing into the vertex, makes the quark-gluon vertex look like the electron-photon vertex of QED with an additional multiplicative factor Ta. The Feynman rules for the QCD vertices can be seen in figure 1.1. The Feynman rules allow the construction of Green’s functions in momentum space. These are the vacuum expectation values of the time ordered products of fields: (27")46(P1~Pn)Gal...an(P11mapn) = flfd4xt.6_’p’x’-(0|Tl¢a1($1)---¢an($n)l|0), 1:1 (1.12) where the space, time and spin indices have all been absorbed into the 0’s. Green’s functions contain all of the physical information of the theory. Green’s functions are used to construct the Scattering matrix (S-matrix), thus the S-matrix will also have all of the physical information contained in the theory. Gar-an (p1, ..., pn) is just the sum of all of the Feyman diagrams contributing to the process of interest. 1.3.2' The S-Matrix and Cross Sections. Green’s functions are not always physical observables: there is no way of guaranteeing that the external particles are on mass shell. In addition, Green’s functions need not be gauge invariant. A relationship exists between Green’s Functions and observables, such as cross sections. Green’s functions are related to observables through the S- matrix. It is helpful to consider a generic toy model using the scalar fields 450 and a coupling 9. A two point Green’s Function Gag has a pole at p2 — m2. Near this pole it has the form of a free propagating field times a scalar constant R4,: (305(1)) -—> R¢Ga5(p)free + finite terms. (1.13) If the particles are hadrons then 12¢ and the physical mass, M, cannot be calculated using perturbation theory. If instead of hadrons we consider the perturbative S-matrix for quarks, then R], and M can be calculated pertubatively in the coupling 9: R¢ = 1+ C(92) and M = m + 0(92). (1.14) We now wish to make the connection between Green’s functions and observables. The connection between Green’s functions and observables is the S-matrx. The S- matrix tells us the amplitude for the scattering of incoming momentum eigenstates into outgoing momentum eigenstates. The most important S-matrix for QCD is the matrix describing 2 —> 2 processes. The S-matrix is derived from the Green’s functions via reduction formulas which relate 005 —> S. The general form for the reduction formula is Gaifl, (ptlf’ee 5((191: 81) + (P2, 32) -* (123,33) + ---(m. 311)) = I111)(p,-,s,-)a, 1/2 1 12¢ XGfl1...,3n(plip2: —p3a _ p”), (1.15) where all of the quantum numbers, for example, spin of the particle 2', are absorbed into the 35’s. 112(p,,s,-)a represents the wave function of the external particle 2' . (101.51. (p;)f"ee is the free propagator for the field 2'. After multipling by 0;; one can set all of the pi’s on mass-shell (19,2 = mg). From the S-matrix the Transition Matrix (T—matrix) is defined as: S = I + z‘T, (1.16) where I is the identity matrix in the space of momentum eigenstates. In the case of momentum eigenstates the T-matrix contains explicit momentum conserving delta functions. These delta functions can be separated from the rest of the T-matrix: ”((191181) + 002,82) ~> (173153)+m(pn13n)) = (270454091 + P2 - p3--- — Pn) > (103,33) + ---(Pn13n))- (1-17) The cross section can be found by integrating the differential cross section which is a function of the M -matrix over the n-particle phase-space (100191.81) + (192.82) —+ (P3133)+---(Pn13n)) = dPSn 4t/(p1.p2)2 _ m¥m§ X |M((pi13i) + (P2182) -+ (193,33) + ...(pn,sn))|2, (1.18) where £13191 4 4 7’ dPSn —-— 1;I WN2(27I) 6 (p1 +192 - Jgpj). (1.19) The Ni’s depend on the normalisation of the wave functions; if mp, s)u(p, s) = 2m then N,- = 1 for vector, scalar and fermions. If U(p, s)u(p, s) = 1 then N; = 2m for fermions. [In the next section we will discuss the treatment of divergences and how they effect the calculation of observable quantities. 1.3.3 Divergences and Renormalisation Schemes Until now we have assumed that the Green’s functions, and therefore observables, were free of divergent terms. This is not true when processes with loop diagrams are included. The inclusion of loop diagrams in the un—modified Green’s functions lead to ultraviolet divergences. Loop diagrams are associated with virtual states in which energy conservation is violated by an arbitrarily large amount. The momentum is conserved at each vertex of a Feyman diagram. The loop momentum, is unrestricted. The 100p momentum, k, is not observable so we need to sum over all possible values. This introduces a f d4k in loop diagrams/ Green’s functions. These integrals over the 100p momentum are often divergent. There are renormalisation schemes that can be used within perturbation theory that remove these divergences. As an example we consider a scalar field for which the un-renormalised loop integral (loop momentum k) is given by: an d4k 1 F (p) (2704 (k2 —m2>((p— k)2 —m2) (”0) 1 d4k 1 = [0 dx/ (27r)4 (k2 — 2xp.k + mp2 — m2)2 (121) 1 d4k 1 = [0 dx/ (271')4 (k2 + 23(1 - 23);)2 — m2)? (122) Going from 1.20 to 1.21 is referred to as Feynman parameterisation. There is a change of variable in the last line k’ = k — :cp. The integral is still divergent in the limit I: —-) 00 (the ultraviolet region) in the present form. The divergence can be understood‘by considering a generic one loop integral: an __ d4k 1 F 0”: mew-Map»? (123) where k is the loop momentum and we have absorbed all of the external momentum dependence into M (p) This integral is undefined due to a logarithmic divergence at infinity. To maintain simplicity the momentum dependance of the Dirac traces and vector indices in the numerator are neglected as they do not eflect the renor- malisation. Logarithmically divergent integrals can be evaluated using dimensional regularisation. In dimensional regularisation the UV loop divergences are regulated by reducing the number of space-time dimensions to n < 4: d4k 2. d4’2‘k W " (“I —<2t>4-2~ (”4’ where e = 2 — n/ 2. The renormalisation scale, a, preserves the dimensions of the couplings and the fields. Within this regularisation method loop integrals like 1.23 lead to poles at e = 0. The minimal subtraction renormalisation prescription (see below) is to subtract off the poles and replace the bare coupling by the renormalised coupling 9012). This leads to expressions such as: - 2 = fibula—gm), (1.25) Plume) —> drew, u) where a mass scale it is introduced which is not present in the original QCD La- grangian. So far there is nothing preventing the mass scale from differing between integrals. We must determine a set of rules to define it for each divergent diagram. The choice of rules used are referred to as the renormalisation scheme. There are two common sets of rules or schemes: 0 Momentum subtraction scheme: choose r(’€")(P0) = 0, (1.26) where the P0 are a fixed set external momenta, and F a specific divergent vertex function. This approach is often used in QED. In this scheme all of the one loop and higher order corrections to the electron-photon vertex go to zero as the momentum transfer goes to zero. P0 is where the photon momentum goes to zero and the electrons are on shell. 0 Minimal subtraction scheme: [1 is chosen to be the same for all divergent inte- grals and is left as free parameter in the renormalised Green’s function. In this scheme )1 will be present in all physical observables calculated from the Green’s functions/M—matrix at any fixed order in perturbation theory. This approach is typically used in pQCD. 10 When the measured inclusive jet cross section is compared to the NLO pQCD pre- diction we will make a choice of which value of u to use, introducing an unavoidable uncertainty in the comparison to data to N LO pQCD. 1.3.4 Renormalisation Scale and Experimental Results Using the minimal subtraction scheme we are left with an arbitrary parameter, )1, in the theory, requiring a method to determine unique experimental predictions from this theory. If we take the simple case of massless particle and a single coupling constant 9, we can compute a cross section, a, from the renormalised perturbative series A 0(1), M) = :1 an(p, (1)9002", (1-27) where A is the highest order term that can be computed. By measuring 0(p, u) for some momenta p0 and fixing )1 to some value we can solve for 9(a). Having solved for g, the cross section a can be computed for any p. Since the cross section, 0(1), p), is an observable it must be independent of the choice of )1: 112153009, )1) = 0. (1.28) This is exact if we perform the computation using all orders of perturbation theory. By computing 0(p, u) to finite order we introduce errors of the order of the first uncomputed term in the perturbation expansion. Provided the coupling is small, leading order or next to leading order should provide a reasonable description of measured observables. The size of the coupling and therefore the applicability of perturbative QCD is closely is related to asymptotic freedom. 11 1 .4 Asymptotic Freedom The use of QCD to describe the strong interaction is underpinned by two prOperties: asymptotic freedom and confinement. We consider hadron spectra to see why these two properties are very important in the success of QCD. Hadron spectra can be described by quark models even though quarks have never been seen in isolation. Although quarks are produced in high energy physics experiments, they hadronize before being detected. Mesons and baryons and / or their decay products are detected. Fiom the hadronization time one can see that the forces between quarks are strong. However, certain high energy cross sections are well described by models in which the quarks do not interact at all (the parton model). Asymptotic freedom refers to the weakness of the forces between quarks at short distances. Confinement arises from the strength of the forces at large distances. QCD allows for both of these behaviours by making the force between quarks a function of the distance. At some distance it becomes easier to create new quark and anti-quark pairs which combine into hadrons than it is to continue to work against the increasing force. 1.4.1 Forces in QCD Asymptotic freedom and confinement arise from the effective forces that are implicit in the Feynman rules of QCD. It is easier to discuss force and potential in the context of QED as it provides insight to QCD processes in a simpler environment. There are differences between QED and QCD: the QCD Lagrangain field tensor has a different structure from the QED analogue. QED has no term for the photon-photon inter- action. However in QCD the gluons couple through the colour charge resulting in a gluon-gluon interaction term. For QED we have: 12 A” —> A” + (1/e)8#oz (1.29) and for QCD we have Gfi —-> 0?, — (1/g)8,)aa — CabcabGfi. (1.30) The additional term in G2} means the field strength tensor has a more complicated form than the QED counterpart. For QCD the field strength tensor has the form: 07,, = ape: — 6.0;: — gfabcszog, (1.31) the QED field strength tensor has no term of the form gfachzGfi. The additional term in the QCD Lagrangian means the kinetic energy term is now not purely kinetic but includes a self interaction be- tween the gauge bosons (gluons). The electrostatic Coulomb force is derived from the potential between two charged particles; V(Q1,Q2,r) — ‘l—QIQ2 — 471 I1" , (1.32) where Q1 and Q2 represent the size of the two charges separated by a vector r. In QED this potential arises from the scattering of two heavy charged particles. If the particles are sufficiently heavy the energy transfer can be ignored as it is much smaller than the momentum transfer (in the non relativistic approximation (122/2M < M )) The potential is the spatial fourier transform of the gauge field propagator, multiplied by the coupling constant at the vertices and divided by —i. If the charges are equal Q1 = Q2 = e the potential is: 13 d3k . 1 _ 2 —k. V“) "- ‘8 f—(271)3e ' ”:1: 2 00 sin(k|r|) 2 — k—. 1. ‘3 (2t)? 0 km ( 33) The lowest order Feynman diagram (tree level, figure 1.2 b) yields the potential. Even beyond tree level the potential is still the fourier transform of the scattering amplitude: d3k V(r) = / We'm’Mkz). (1.34) In the previous example A(k2) is given by single photon exchange ~ 0(e2). Fig- ure 1.2 c, d, e, f, 9 show the 0(e4) graphs that contribute to the potential through perturbative corrections. The momentum dependence of the different contributions may differ from the lowest order term. These higher order diagrams require renor- malization but it is instructive to assume it is done and consider the overall physical picture. Experimentally, the contributions of various diagrams from the lowest order di- agram cannot be separated. The higher order corrections modify the momentum dependence, and therefore the potential. To determine the electromagnetic coupling we define the amplitude at a fixed momentum transfer —k2 = p0 to be A(po) = ——-—a(p0), (1.35) PO where the fine structure constant is equal to 62 = Z? (1.36) Cl This tells us nothing about the momentum dependence of A(p). The main contri- bution of the higher order diagrams comes from the process where the two incoming 14 charges are linked by a virtual photon inducing a self energy diagram of a fermion and anti-fermion pair. The net charge of the fermion/anti-fermion pair is zero and they act to screen each of the original charges from each other. We can think of the two heavy charges as being surrounded by a cloud of charge pairs. If the heavy charges are far apart they each see a large cloud which serves to decrease the effective charge of the other heavy charge. As p0 increases the charges come closer together (uncertainty principle) and once inside the cloud the screening is less effective. This can be summarised as a statement, that, as the momentum transfer increases, the observed charge also increases: d 2 —e > 0. 1.37 dpo ( ) We define the effective charge for QCD as g2(p0) and the effective fine structure ’constant’ for QCD by as = —. (1.38) The diagrams for QED are all present for QCD also, with photons replaced with gluons. There are additional diagrams due to three gluon vertices. As in QED the effect of the virtual corrections is to surround the now non-abelian charges with a charge cloud. The emission of a gluon does not leave the charge of the heavy charge unchanged. The total charge is still conserved but the charge of the heavy scatterer leaks into the surrounding charge cloud. As the two scatterers enter into each others virtual charge cloud they are less likely to see the true charge of each other [5]. This is the opposite of the QED situation. Here we have ’anti—screening’: — 2 < 0, (1.39) so as p0 increases, the observed coupling decreases, giving asymptotic freedom (larger momentum transfer smaller coupling). Large momentum transfer corresponds to small distance scale, and therefore small coupling, which aids in the use/validity of the perturbative description. 1.4.2 The Renormalization Group and the Effective Cou- pling To define A(p0) in perturbation theory it is necessary to introduce a renormalisation mass. This can be done via the inclusion of ,u in A(p0) = egg—Q). In terms of a3(u2), the amplitude is of the form “103/#2) 2 2 k2 + 020053 408) = astuzrfig + 612103042) of”) + (1.40) with an a number and 0.20 possibly a function of the masses and the infrared cutoff. The renormalisation group consists of the set of all possible rescalings of u. The amplitude A(k2) is a physical quantity and can therefore be measured experimentally. For this reason the amplitude cannot depend on the choice of 112: dip/109)] _ T _ 0. (1.41) Using equations 1.40 and 1.41 we see the u dependence is described by: 2 19% = —a21a§(112) + higher order terms. (1.42) Given an > 0 the coupling decreases as the renormalisation scale it increases. The asymptotic freedom can be expressed through the dependence of the linear coupling 9(a) = (/47ras(u2), which yields: 16 u— = M900), (1.43) where B is the power series: [3(9) = -9(%}’-51 + (EVE + ..). (1-44) 61 can be derived from am or from any physical quantity that depends on u in perturbation theory. For QCD 51 =11 —2nf/3= (11Nc—2nf)/3, (1.45) where nf is the number of flavours of quarks and NC is the number of colours. The lowest order approximation to the dependence on the linear coupling in terms of 0:2 is c1803): “3%; 2 2 (1.46) 1+ (31/4W)as((40)1n(# mo) or equivalently 2 _ 47? “8‘“ )‘ straw/A2) ”'47) where A : ”Ugh/(magma), (1.48) this sets the scale for the running coupling otherwise known as AQCD- By using higher terms in the ,8 power series a more accurate result can be obtained. 023012) can be expressed in an expansion of powers of 17 Haw/42>), (1.49) where the coefficients of the 1/(ln(u2/A2)) series is a polynomial in ln(1/(ln(u2/A2))). Keeping the first two B terms allows the determination of coeffecient of the [1 / (ln(,112/x\2))]2 term: c.3042) __ 1 62ln(ln(u2/1\2)) 1 4w" ‘mznofl/Ae" straw/A2) ”(Wm/42)” “'50) where )82 = 102 — 38n f / 3. We have set a renormalisation scale by the introduction of a unit of mass )1. 1 .5 Infrared Safety Using the solution to the running coupling discussed above we see asymptotic freedom can aid one in practical cases. For a given physical quantity 0(1),- .p,- / 112, m? / 112, 9(a)) that can be calculated in perturbation theory. It follows that: 00 2 . . p..p. m. 2 1 2 2901)) = 2: an( 3 2]! 5 )01310‘)’ (1'51) u ”:0 I! u where the p,- denote the external momenta and m,- are the internal (quark) masses. It is not uncommon for the coefficients on to be large, regardless of the value of of a3(u). Many cross sections in pQCD are infrared divergent due to the vanishing gluon mass. Infrared divergences either cancel or can be factorised into universal functions such as Parton Distribution Functions, which contain all of the low energy information. Infrared safe quantities are those that do not depend on the long distance behaviour of the theory. For the class of quantities that are infrared safe the coefficients an are infrared finite [17, 18] and also have a finite limit for vanishing mi: 18 . . m2 2 m2 odfifl-p—igo»=a(%,o,g(m>{1+0(-Q—g)}, (1.52) Q2 is the scale of the large invariants alon the p-.p -. For an infrared safe quantity 3 1 J equation 1.28 has the solution (upto mass corrections ~ m2/Q2) . . m2 06;? , 771.9(2)) = «103(2)» (1.53) all the momentum has been absorbed into the couplings. If Q is large the coupling decreases and the description offered by pQCD should impove. 19 a) the b) mm 11.1 BJ b,k'.-" c,k V2829, v3ragrp3 v,,a,,p, Vvavm Figure 1.1: Feynman diagrams/rules for the vertices. All momenta defined to flow into vertex. a) — saint-1111113.. (1.57) vacated. (1.53) C) - 901116263 [9”2 (p1 - 292)"3 + 9"?"3(p2 - 193)"1 + 9"3"1(p3 - p1)”"’l (1-59) d) — z.92ICealaQC'ea3a4(gt/1189112”4 - 9”“’“9"2"3)l +("Z'92l[0601613Ceataze(gull/"91’3"2 — gull/29‘3”” +(‘igzl [Cea1a4 Cea2a3 (gull/2.9M“,3 - 9V1V39V4V2 )] (1.60) 20 Figure 1.2: a) An ultraviolet divergent one loop scalar diagram. b) Lowest order potential QED/QCD ( tree level diagram ). (c-e) Field theory corections to the potential in QCD which are also present in QED. (f-g) non abelian correction to the QCD potential ( not present in QED}. 21 a,i BI] a ---------- > ........... b V v, utb Figure 1.3: Feynman rules for a) quark propagator, b) ghost propagator and c} gluon propagator. a) 2' ,2 if; +1.” + must b) ikzéz-a is c) (Covariant gauge) ik—gfikgw + (1 — ~:—)k];“::€] c) (Physical gauge) 2' k2 if“ [—g’“’ + kflnanknflkV — n2 (:”:;2] 22 (1.65) (1.66) (1.67) (1.68) 1.6 Jet Production As discussed previously one of the most important QCD processes in pp collisions is the leading order process of 2 —> 2 scattering. Two incoming partons interact and produce two outgoing partons. These outgoing partons hadronize to yield a jet of hadrons. The particles that make up the hadron jet deposit energy in the calorimeter. These energy deposits are what we measure (calorimeter jets). Figure 1.4 show the diagrams that contribute to jet production. All other diagrams that contribute to lowest order jet production are related to those shown via crossing. At this point it is usefull to define some of the variables that will be used through- out the following work: ET and PT are the transverse energy and momentum of the partons/ jets. In the limit of massless partons/ jets ET = PT. The rapidity of jets and partons is defined as: Y = %ln (gég). The pseudo rapidity is defined as: 17 = -ln(cot(0/2)). In the massless limit Y = r). The two-jet cross section can be expressed in terms of the parton distribution functions and a matrix element M [7]: (130 = 1 2 £031,112) fi($2, #2) dY3dY4dP12. 16t32ijk,=qqg :31 :12 1 x M “—th 2 , 1.69 ZI (v “1+6“ ( ) where PT is the transverse momentum, the f, represent the parton distribution func- tions and Y3 and Y4 are the rapidities of the outgoing partons. 2:1 and 2:2 are the momentum fractions 1/2:1:T(eY3 + eY4) and 1/2$T(e‘Y3 + e’Y“) respectively, where :rT = 2pT/\/§. In the parton center of mass frame the subprocess scattering angle, 0*, is related to the lab frame rapidity difference: 23 1 W' (1.70) Y* = %(Y1 — Y2) by sind" 2 Table 1.1 shows the square of the invariant matrix element (|M l2) for 2 —-> 2 parton subprocesses contributing to jets. The value is given in terms of the Mandelstam variables: §,t and it. § — (Pi + Pj)2 A 1 t = (P,- — P1)2= —§s s(1 — cosfl") a _ (P,- — P1)2— _ —-;-s (1 + cosfl"), (1.71) where p,- and pj are the initial state four-vectors and p1 is an outgoing momentum four-vector. The Mandelstam variables obey the relation 3‘ + t+ i1 = 0. If we assume we have massless quarks and perfect jet algorithms (P Jet = Pparton) the single jet cross section can be found by integrating equation 1.69 over one of the jets: E- d3a d1: d2: ———]et = [11 2f( (a: 2 172 3 , 2 t 1#2)fj($21fl) (-) d PJet 167r s ',j,klE:_ _qq,g 0 1:1 3:2 M"-—>kl2 6“+t+”. X XI (2.7 )I 1 l (Ski (5 u) Predictions for the jet cross section as a function of PT are obtained from the expression Ejetd30 = d3o _> 1 d20 d313,; " dQPTdY 27rPT dPTdY’ (1.73) where the last term arises from the assumption that jets and partons are massless (ET 2 PT and r} = Y). 24 Up to now our discussion of jet production was based on exclusive dijet production. Typical events seen at CDF have more than 2 jets. To understand the multijet final states we need to consider initial state radiation, final state radiation and higher order diagrams. The basic 2 —+ 2 interaction, or hard scatter, is a QCD interaction. In QCD however, the quarks and gluons can radiate gluons in both before and after the hard scatter. There can also be extra partons produced in the hard scatter, when this occurs the process is no longer 2 -—> 2 but is now 2 -—> 3, 2 —+ 4, etc. Experimentally, the inclusive jet cross section is defined as the number of jets in a bin of PT, normalized by the acceptance and integrated luminosity. As an inclusive quantity all the jets in each event which fall within the acceptance region contribute to the cross section measurement. 25 (a) i 3X >>(b) v (99, (C) Figure 1.4: Diagrams that contribute to jet production. All other lowest order jet production diagrams are related to this set by crossing. The low PT cross section is dominated by qg and gg. The high PT cross section is dominated by contributions from the qg and qq subprocesses. The Figure is taken from [8] 26 pp --> jet +X «ls = 1800 GeV CTEQSM 11 = E, /2 0< lnl <5 1 T I I ' I ' I I I I j I ' I —qq —--99 Subprooess fraction 4 L 400 450 500 150 200 250 300 350 ET (GeV) 0 1 I 50 1 00 Figure 1.5: QCD subprocess contributions. The figure describes the contributions for the subprocesses gg, gq and qq at J5 = 1.8 Te V. We expect very similar contributions at 1.96 Te V. 27 Process 2 IMIZ/g4 9* = 7r/2 99’ -+ 99’ 3&2??? 2’22 99’ —+ 99" 3332312 2'22 qq -> 99 afigfi + £32213) " £72317: 3'26 99 -+ 9’9’ 3933'? 0'22 q'g __) qg 3(523‘2 + 52:29?) _ %%.:. 2.59 <19 —+ 99 3%éit‘2f‘2- 3932a? 1'04 99 —> 99 $2232" 3932'? 0'15 99 —+ 99 ‘38??? + "2;? 6'11 99-+99 §(%—%‘%‘% 30'4 Table 1.1: Leading order jet production matrix elements squared (Z IMP/94). The spin and colour indices have been averaged (summed) over the initial (final) sates. The column labelled 0* = 7r/2 gives the size of the contribution from each of the subprocesses at 0* = 1r/ 2 . Table adapted from [7, 12]. 28 Chapter 2 Jet Identification 2.1 Introduction In this chapter we give a brief history of jet measurement and begin to link the theoretical concepts of QCD to the application of QCD as a predictive tool to be used in an experimental setting. Searches for jet structure at the ISR pp collider (\fs' = 63 GeV), provided hints of two jet signatures. Extraction of the jet signal was difficult because the sharing of hadron momentum between the constituent partons reduced the available energy for parton scattering [13, 14, 15, 16]. In addition to the low jet energies, the remnants of the incident hadrons were a large background of low energy particles, another factor making jet identification difficult. The first clear evidence of two jet dominance was seen at the CERN Spp'S collider (\/E = 540 GeV) [27, 21]. This was also the first measurement of the inclusive jet cross section. Increases in the center of mass energies and improvements in accelerator/ detector technology gave rise to larger sample sizes and increased collision energies. These improvements lead to production of jets of higher transverse energy. Producing jets at higher energy helps to distinguish jets from the underlying event (beam remnants from initial hadrons). The higher energies also reduced the transverse spreading of the 29 jets in space during fragmentation. Figures 2.1 and 2.2 show a dijet event resulting from 3. pp collision with \/5 = 1.96 TeV, seen at CDF. The figures illustrate very well the separation of the jet signal from the background and the 2 jet dominance of pp collisions with a large center of mass energy. -0 ’I , 9” ‘ O I ’ ’ O _ - r r o ,fo,:,,:;,:. 0 I , o o '9 a, Figure 2.1: Lego Plot of CDF dijet event. The display shows two well separated jets in the central calorimeter. The pink and blue colouring of the towers indicates the fraction of electromagnetic energy and hadronic energy respectively. Figure 2.2: Central outer tracker (COT) and calorimeter display of a CDF dijet event. The display highlights the back to back nature of the dijet event. COT superlayer hits and tracks are also seen in the display. 30 Along with improvements in high energy experiments, there has also been progress towards more detailed and precise theoretical predictions. As the transverse energy of the jets increase, the value of the strong coupling constant as decreases, improving the validity of the perturbative expansion. At leading order, 0(03), one parton from each incident hadron participates in the collision producing two outgoing partons. More than two jets are observed in a typical collision at the Tevatron. To account for the multi-jet contributions, leading log Monte Carlo programs were developed to take the leading order Matrix Element predictions and add parton showering. The additional showering transition from partons to hadrons was based on empirical models of hadronisation and fragmentation and allowed for the description of multi-jet final states. The cross section for hard scattering between two incident hadrons (1 + 2 —> 3 + X) to produce hadronic jets can be factorised into components from empirically ' determined Parton Distribution Functions (PDF’s), f,(:z:, [1%) and the perturbatively calculated two-body scattering cross section a. A detailed discussion can be found in [22]. The hadronic cross section can be written as (using u 2 pi, = pf, where p,- is the renormalisation scale) 01+2—>3+X = 3233' /d$1d$2fi($1,Hzlfj($2:#2) X 5i,j($1P: $213, 03012»- (2-1) The PDF’s, f,(.v, uz), describe the initial parton momentum as a fraction 9: of the incident hadron momentum P and a function of the factorisation scale up. The index refers to the type of parton (gluon or quark). The relative contribution of a given sub-process is shown in figure 1.6. The PDF’s are universal and can be derived from any process, e. g. Drell Yan, and applied to any other process. The PDF’s are derived from global fits to scattering data 31 taken from a variety of experiments measuring different processes. Uncertainties in PDF’s arise from uncertainty in the input data, the parameterisations of the parton momentum distributions and the extrapolation of the PDF’s into other kinematic regions. The hard two body parton level cross section, a, is only a function of the fractional momentum carried by the incident partons (:r), the strong coupling constant (as) and the renormalisation scale that characterises the energy of the hard interaction (u). The two body cross sections can be calculated with perturbative QCD at leading order (L0) [24] and next to leading order (NLO) [23, 25]. At leading order there are eight diagrams that describe the 2 -—> 2 scattering. The NLO calculations include diagrams with gluon emission as both an internal loop and final state parton. The scales u R and p F are uncertainties which are intrinsic in fixed order per- turbation theory. Although the choice of u scale is arbitrary, a reasonable choice is related to a physical observable such as the jet PT. In the following analysis we will compare the inclusive jet data to NLO QCD predictions. Predictions for the jet cross section as a function of PT are obtained from the general cross section given above: Ed3o = d3o d3p _ d2PTdY 2 = 1 do . (2.2) QWPTdPTdY Experimentally the inclusive jet cross section is defined as the number of jets in an PT bin normalised by acceptance and integrated luminosity. The fundamental step in the measurement of the inclusive jet cross section is the identification of jets. We need to be able to identify jets in a consistent way at 32 parton, hadron and calorimeter level. The parton level jet identification is required for comparison of data to theory. In a leading order (LO) calculation there are two partons in the final state, each of which will be associated with a jet. LO predictions have no dependence on the jet algorithm or on jet shape or size. When considering next to leading order (N LO) calculations there can be up to 3 partons in the final state and therefore there can be more than two jets in an event. The requirement to find jets in a consistent manner at parton, hadron and calorimeter level can be satisfied by the cone algorithms J etClu and MidPoint and also in the KT clustering algorithm. There are subtle differences between the clustering scheme used at parton level and calorimeter level. At next to leading order parton level there are no overlapping jets, but at calorimeter level we may see multiple jets that do overlap. Jets that overlap are either split into 2 (or more) non-overlapping jets or merged into a single jet. . The splitting and merging procedure depends on the algorithm. This splitting and [merging feature is modelled in the parton level clustering by a parameter Rsep. Like u, we will see that when comparing data to NLO pQCD we will need to make a choice for the value of Rsep to use. The details of the algorithms used in the current analysis are given in the following chapter. 33 Chapter 3 Jet Algorithms 3.1 Introduction In the present analysis there are two cone algorithms used to reconstruct jets: Mid- point and JetClu. The JetClu algorithm is used in the Level-3 trigger and the Mid- point algorithm is used for jet reconstruction for the inclusive cross section measure- ment. Cone algorithms form jets by grouping together particles whose trajectories, or towers, lie within a circle of radius R in (n(Y), (b) space, where 17(Y) is the pseudora— pidity(rapidity) and (b is the azimuthal angle. How the towers are combined and how the jet properties are calculated depend on the specific algorithm. In this section the ideal theoretical and experimental attributes of jet algorithms are outlined, followed by the outline of the recombination schemes used in the JetClu and Midpoint algorithms. 3.2 Theoretical Attributes of a Jet Algorithm There are some desirable attributes we look for in jet clustering algorithms. These attributes may not be be present in an ideal form in practice due to the practicality of the implemention and /or computing limitations. Some desirable features are: 34 0 Infrared safety: A jet algorithm should not only be infrared safe in the sense that infrared singularities do not appear in any perturbative calculations but it should also cluster partons, hadrons and calorimeter towers in a manner which is insensitive to soft radiation. 0 Collinear safety: The algorithm needs to be collinear safe with respect to per- tubative calculations and also find jets that are insensitive to collinear emission of radiation. 0 Invariance under boosts: The algorithm should find the same solutions inde- pendent of boosts in the longitudinal (beam) direction. This is important in pp situation where the center of mass of the individual parton-parton interaction may be boosted with respect to the pp center of mass. 0 Order Independence: The algorithm should find the same jets at parton, hadron and calorimeter level. 3.3 Experimental Attributes of a Jet Algorithm After a jet has passed through the calorimeter the effects of showering, detector response, noise and multiple interactions will affect the performance of any jet algo- rithm. It is our goal to remove these effects and correct the measure cross section to the hadron and parton levels (removing detector effects). To aid in the correcting of the calorimeter level cross section to the hadron and parton levels the following attributes are desirable: o The algorithm should not be detector dependent. There is a need to avoid or minimize any dependence of the algorithm on the segmentation of the calorime- ter used to take the data. 35 0 Resolution and angle bias: The algorithm should not amplify the effects of resolution smearing of the detector. Minimising this also helps to limit the size of the corrections that will be required to go from the calorimeter level cross section to the hadron level cross section. 0 Stability with luminosity: The jet finding efficiency should not by strongly affected by multiple hard interactions associated with high luminosity. Also, the jet energy and angular resolution should not depend on luminosity. 0 Fully specified: the algorithm should include specifications for clustering, energy and angle for reconstructed jets. In the case of overlapping jets the splitting and merging criteria need to be specified as well. 3.4 JetClu The JetClu algorithm is used in the Level-3 trigger (discussed later) to reconstruct jets from energy deposits in the calorimeter towers. The algorithm consists of preclus- tering, clustering, splitting and merging, and the calculation of jet parameters. When used in the Level-3 trigger the tower energies are not corrected for the primary vertex Z. Below we outline the clustering steps. 3.4.1 PreClustering 0 Merge the towers in the Foward and Plug calorimeters such that they have 24 segments in (b. 0 Make ET ordered list of towers with ET > 1GeV. 0 Associate the highest ET tower with the first precluster. 0 Loop over the tower list and add a tower to a precluster if it is within 7 x 7 towers of the seed tower and is adjacent to an existing tower in the precluster. 36 Otherwise, start a new precluster. 0 Restore full segmentation of towers in the foward and plug calorimeters. At this point a precluster consists of a contiguous set of towers with decreasing energy. Every tower with ET > 1 GeV is assigned to one and only one precluster. 3.4.2 Clustering 0 Make list of towers with ET > 100 MeV. 0 Order preclusters in ET. 0 Add tower to a precluster if the tower is within AR 2 0.7 of the cluster centroid. o Iterate until tower list of the clusters is stable. 0 Original towers in cluster are never dropped . At this stage every precluster has an associated cluster. A tower may belong to more than one cluster. 3.4.3 Merging and Splitting 0 ET order the clusters. 0 Using a double nested loop ( i = 1,number of clusters, j = 1,i — 1), make a list of towers which are included in both clusters i and j. 0 Merge two clusters if the common towers contain more than 75% of the ET of the smaller cluster. 0 If overlap contains less than 75% assign common towers to the closest cluster in an iterative fashion. 37 At this stage a tower is assigned to one and only one cluster, all clusters with ET > 1GeV are promoted to jets. The final jet energy and momentum is computed from the final list of towers: EJet Z Ei 2 Pa; = ZEisinw )cos (obi) Py = ZEisinwi )sin (43,-) P2 = ZEz-cos(0,) i m... = mark?!) 3 ,/P3 + P3 3in(0Jet) = \/P3 + P3 + P],2 EJet = E etsin 9 Ct) 3.1) T J J 3.5 The Midpoint Jet Algorithm In this analysis the Midpoint jet algorithm is used in the reconstruction of jets at the hadron and calorimeter level in Monte Carlo, and at calorimeter level in the data. The Midpoint algorithm has some advantages over the JetClu. Unlike JetClu, Midpoint does not use the calorimeter segmentation when clustering at the hadron level. This is very important as we use the hadron level clustering when deriving jet corrections. Both JetClu and Midpoint are seed based algorithms, they look for jets only around seed towers, which can lead to sensitivity to soft radiation. The Midpoint algorithm places additional seeds at the midpoint positions of stable cones: 38 Pi + Pj, P,- + P, + Pk etc. These additional seeds at the midpoints are used to define new initial search cones. The addition of these search cones lessens the sensitivity of the algorithm to soft radiation. 3.5.1 Clustering The MidPoint algorithm makes use of 4-vectors throughout the clustering. The de- tector towers are sorted in descending PT. Only towers passing a seed cut, P}: "we" > PTseed are used as starting points for the initial jet cones. The seed threshold is choosen to be low enough such that variations of P156“! lead to negligible variations in any jet observable. A tower or parton i is clustered into a cone and eventually a jet if the separation in (Y, (b) satisfies the following: i c: \/(Yi — Y6)2 + (vi — (to)? g R, (3.2) where c denotes the cone variables. For massless towers, particles or partons Y = 17. The centroid corresponding to this cone is given by PC = (EC,PC)=Z(E",P;;,P;,P§) iCc 1 EC+PC Y6 = §ln(EC—P:C) PC (1)6 = tan-1(P—gé). (3.3) 1' A jet arises from a stable cone, for which 170 = Y0 = Yjet and $6 = dc = obj“, and the jet has the following kinematic properties: Pjet : (Ejet,Pjet)= Z (Ei,P;,P;,P:) iCJ=C Wet : 11n(Ejet + Pget Ejet _ Pget 2 ) 39 (bjet = tan—1(Py. ). (34) 3.5.2 Splitting and Merging The Midpoint algorithm has a different splitting and merging criteria from JetClu. In the MidPoint algorithm two jets are merged if the overlap energy is greater than 50% of the smaller jets energy. This splitting/merging is performed on an iterative basis. 40 Chapter 4 The Detector 4.1 Introduction The inclusive jet cross section is measured from pp collisions in the Tevatron ac- celerator at Fermilab. The final state is measured using the Collider Detector at Fermilab (CDF). This chapter provides a brief description of the accelerator complex at Fermilab and of the subdetectors of CDF that are central in this analysis. 4.2 Experimental apparatus 4.2.1 The Accelerator Complex The pp collisions at Fermilab are made possible by a series of accelerators culminating in the Tevatron. The Tevatron is currently the world’s highest energy accelerator; during the data taking period of this analysis, the Tevatron produced collisions with a center of mass energy of \/§ = 1.96 TeV. 4.2.2 Protons The proton source at Fermilab is composed of a 400 MeV linear accelerator (Linac) and an 8 GeV Booster. The Linac is accompanied by a H " ion source Cockcroft- Walton accelerator (capacitor-diode voltage multiplying array). The process of proton 41 acceleration begins with a bottle of molecular hydrogen. The H - is extracted elec- trostatically using a cesium walled chamber. The molecular hydrogen is ionised due to the low work function of the cesium. A 750 keV electric potential is applied to the resulting ions by a Cockcroft-Walton power supply. The H ' ions are accelerated electrostatically to 750 keV. Following this acceleration they enter a transfer station. The transfer station gives a bunch structure to the now continuous H ‘ beam and injects the bunches into the 150 m Linac. The Linac consists of 11 copper radio frequency (RF) cavities. A potential difference is applied to alternating cavities, this accelerates the H ‘ ions to 400 MeV. At the end of the Linac, a copper foil strips the electrons from the H ' ions leaving a bare proton. The protons are then injected into the Booster. The Booster is an alternating gradient synchrotron with 475 m circumference. The Booster accelerates the protons to 8 GeV. From the Booster the beam is transferred into the Main Injector (MI). In collider mode the MI, which is also a synchrotron, accelerates the proton beam to 150 GeV. It also performs the coalescing and cogging of the beam preparing it for injection to the Tevatron. The Tevatron is a superconducting synchrotron with a circumference of ~ 4 miles. This accelerates the beam to its final energy of 980 GeV. 4.2.3 Antiprotons Antiproton production begins by extracting the 120 GeV proton beam from the MI and directing it onto a nickel target. In the resulting nuclear interactions, antiprotons are produced. The yield is approximately 1 antiproton for every 105 protons that hit the target. The resulting spray of particles is focused by a cylindrical lithium lens with an 0.5 MA pulsed axial current. The particles are then filtered by a pulsed dipole- magnet spectrometer resulting in an 8 GeV beam of antiprotons. The antiproton beam is directed toward the Debuncher, one of two rounded triangular synchrotrons 42 which make up the antiproton source. The Debuncher reduces the momentum spread of the antiproton beam by bunch rotation and stochastic cooling techniques. The cooling process converts narrow bunches with a large momentum spread into a broad beam with a small momentum spread. After the beam is cooled it is injected into the Accumulator, which is co-centric with the Debuncher. From the Accumulator the antiprotons are loaded into the main injector. From here they are loaded into the final stage of the Tevatron. 4.2.4 Collisions For Run II the Tevatron operates with a 36 on 36 bunch structure, with a 396 ns bunch spacing. At two points on the Tevatron ring (BO and D0) the beam is focused using quadrapole magnets to achieve a high luminosity at the interaction points inside the detectors. The luminosity of the beams is given by H E— I [3*ep(1+ 2%) , L = EfONfiNpB (4.1) where 'y is the relativistic energy factor, f0 is the revolution frequency, Np (Nfi) are the number of protons (anti-protons) per bunch, B is the number of bunches of each type, 6* is the beta function at the center of the interaction region, 6p (65) are the proton (anti-proton) 95% normalised emittances and H is the form factor associated with the bunch length. 4.3 The CDF Detector CDF is a general purpose detector located at the BO interaction point of the Tevatron. It is cylindrically symmetric around the beam axis and has back-foward symmetry about the nominal interaction point. It is designed to make precise position, mo- 43 mentum and energy measurements of particles originating from the pp collision. This section describes the Run II configuration of the CDF detector. A more complete description can be found in the technical design report [34]. The components of the detector that are central to this analysis such as the Calorimetry, Cherenkov luminos- ity counters (CLC) and central outer tracker (COT) will be outlined in the following sections. CDF uses a right-handed coordinate system: :3 points away from the center of the Tevatron (north), 9 points upward, and 2 points along the beam direction (east). Due the cylindrical symmetry of the detector it is useful to use cylindical coordinates for the physical quantities used in measurements. Using r, the radial distance from the z axis, (13 is the azimuthal angle (0 radians lies on the :1: axis) and 0 is the polar angle relative to the z axis. The rapidity (Y = §ln(%§£§)) is a relativistic invariant for boosts along the beam axis. In the ultra relativistic regime the rapidity can be approximated by the purely geometric quantity psuedo—rapidity (n = —ln(cotg)). The CDF detector is a combination of tracking systems inside a 1.4 T solenoidal magnetic field surrounded by electromagnetic and hadronic calorimeters and a muon system. The measurement of the inclusive jet cross section uses the calorimeters for measurement of the jet energy/momentum. The tracking system provides the position of the pp collision vertex. This vertex is used in the offline reconstruction of jets. Closest to the beam pipe is the Silicon Vertex Detector (SVX). It is roughly 60cm long and covers the radial region 3.0 cm out to 7.9 cm. The r — d) tracking information is provided by the SVX allows precise determination of the transverse position of the event vertex and contributes to the track momentum resolution. Surrounding the 44 SVX is the Vertex Drift Chamber (VTX). This detector provides r — Z information used to determine the position of the pp in z. The SVX and VTX are inside a 3.2 m long drift chamber called the Central Tracking Chamber (CTC). The CTC covers the radial region from 31.0 cm to 132 cm. The momentum resolution of the SVX-CTC system is (SP/P = [(0.0009PT)2 + (0.0066)2]1/2 where PT has units of GeV/c. Outside the tracking system there is the combination of electromagnetic and hadronic calorimeters. Calorimetery is used to measure the energy of incident par- ticles. The central calorimeters (Inl < 1.1) consist of projective towers of dimension An x A¢ = 0.1 x 15". The inner section is an electromagnetic compartment designed to measure the electromagnetic energy of incident particles. The outer compartment is hadronic. Each tower consists of a unique piece of the solid angle and the calor- metric information within that piece of the solid angle. CDF has several interface regions between calorimetry detectors of varying (b and 17 segmentation. There are regions where the electromagnetic segmentation is finer than the hadronic segmenta- tion. There are nine distinct types of tower: 4.3.1 Central Outer Tracker The Central Outer Tracker (COT) is an open cell drift chamber which provides track- ing coverage for the region [77] < 1. The COT is segmented into 8 layers moving out radially from r = 40 to 137 cm. Each layer holds a number of cells, the cells contain a 50 : 50 mixture of Ar-Et gas and a trace amount isopropyl alcohol. The Ar—Et and isopropyl alcohol combination has a drift velocity of ~ 200um/ns. The maximum drift length for a given cell is approximately 0.9 cm and the maximum drift time ~ 175ns in the drift field of 1.9 kV/cm. For high PT tracks the beam constrained momentum resolution of the COT is (SPF/P]? g 0.001 (GeV/c)”1. 45 4.3.2 Magnetic Field The CDF detector has a 1.4 T axial magnetic field throughout the tracking volume which enables measurements of charge and momentum via the tracking detectors. The field points in the —2 direction of the CDF global coordinate system. The solenoid used to generate the field is superconducting and is constructed of an aluminum stabalised NbTi conductor. The normal operating field of 1.4 T corresponds to a persistent current of 4650 Amps. The cooling of the solenoid is done indirectly using liquid helium. The soleniod is supported by an aluminum structure and an iron return yoke. 4.3.3 Calorimetry The calorimeter systems at CDF surround the tracking volume and the solenoid. They provide the energy measurement of electrons, photons and jets. Each calorimeter system covers Zn in azimuth, and a large range in n. Electromagnetic The central electormagnetic calorimeter (CEM) is a lead-scintillator sampling calorime- ter consisting of a stack of 1 / 8” thick lead plates separated by 5 mm thick polystyrene scintillator. They sample the electromagnetic shower in the regions bounded by the lead plates. Electromagnetic particles interact with the lead causing showering of electrons and photons in the calorimeter. The electrons produce blue light in the scintillators. The total amount of light observed at a photomultipler tube (PMT) is proportional to the energy of the initial electron or photon. The light is collected by acrylic wavelength shifting fibres at both azimuthal tower boundries and guided to the PMT’s. In order to maintain a constant radiation thickness of X0 = 18 (X0 is the radiation length) as a function of n, the layers of lead are replaced with acrylic. 46 At [17] = 0.06 there are 30 layers of lead; at [n] = 1.0 there are 20 layers of lead. Hadronic The Central Hadronic (CHA) and End Wall Hadronic (WHA) calorimeters are made up of layers of 2.5 cm thick steel separated by 1 cm thick plastic scintillator. The central electromagnetic calorimeter (CEM) is followed at larger radius by the the central hadronic calorimeters (CHA and WHA ). The CEM absorber is lead and the CHA/WHA absorber is 4.5 interaction lengths of iron; scintillator is the active medium in both types of calorimeter. Two phototubes bracket each tower in 03 and the geometric mean of the energy of the two tubes is used to determine the 4) position of the energy deposited in the tower. 4.3.4 Resolution of the calorimeters The CEM has an energy resolution of 0(E) _ 14.0% E x/FE where GB indicates addition in quadrature. At a depth of about 6 X0, the CEM ea 2%, (4.2) contains a shower maximum detector called the CES. This employs a proportional strip and wire counters in a fine-grained array to provide precise position and shape information (~ 2 mm) for electromagnetic cascades. The CHA is an iron-sintillator sampling calorimeter, approximately 4.5 interaction lengths in depth and has an energy resolution of o(E) 50.0% E x/FEF e 3%. (4.3) 47 The WHA is also an iron-scintillator sampling calorimeter covering the psuedora— pidity range 0.7 < [1)] < 1.3. Like the CHA the WHA has a depth of ~ 4.5 interaction lengths, however the resolution is somewhat poorer with a resolution of 0(E) _ 75.0% E VET e 3%. (4.4) 4.3.5 Cherenkov Luminosity Counters (CLC) The Cherenkov luminosity counters are used to measure the luminosity at CDF. The counters provide the trigger requirement for the minimum bias data sample used in this analysis. The CLC consists of two conical shaped volumes containing isobutane as a radiator. These volumes are located on both ends of the experiment in the three degree hole between the end plug calorimeter and the beam pipe. Each of the volumes is divided into 48 conically shaped mylar counters that are arranged into three concentric rings around the beam pipe. The CLC counters accept particles from the collision point in the psuedo-rapidity range 3.7 < [7)] < 4.7. The CLC samples a large fraction of the total inelastic cross section. The min bias trigger is based on coincidence triggers from the east and west CLC modules using a 15ns time window centered on the bunch crossing time t — 20 ns.. 4.3.6 Segmentation Name Rapidity gb — r] segmentation CEM 0.0—1.1 CHA 0.0—0.9 15" x 0.1 WHA 0.7-1.3 Table 4.1: Segmentation of the central calorimeters 48 Chapter 5 The CDF Trigger 5.1 Introduction The trigger plays an important role at CDF as the collision rate is much higher than the rate at which data can be stored and many of the collisions do not contain interesting physics. The role of the trigger is to extract the most interesting physics events from a large number of minimum bias events (see chapter 6). The trigger can be used to preferentially select high transverse energy jet events while rejecting the more numerous minimum bias, or zero bias events. This allows us to obtain a large number of events covering a large jet ET range without saturating the bandwidth with uninteresting events or low ET jets. In this chapter we give an overview of the CDF trigger system and describe the jet triggers used in the analysis. A more complete description of the trigger system can be found in the Technical Design Report [34]. 5.2 The CDF Trigger Architecture The CDF trigger has a three level architecture with each level providing a rate re- duction large enough to allow processing at the next trigger stage, the reduction is achieved by requiring an event has specific properties at each trigger stage. The Level-1 trigger uses hardware to find physics objects based on a subset of the avail- 49 able detector information. The Level-2 trigger uses hardware to do limited event reconstruction in programmable processors. The Level-3 trigger uses the full detector information to reconstruct events in a processor farm. 5.2.1 The Level 1 Trigger The Level-1 hardware finds calorimeter objects, and tracks in the central tracking chamber. The decision on whether the event satisfies a trigger is made every 132ns. Level 1 Calorimeter Hardware The Level-1 calorimeter hardware triggers on electrons, photons, total event trans- verse energy, missing transverse energy, and jets. The calorimeter triggers are divided into two types: object triggers (jets, electrons and photons) and global triggers ( SET and missing transverse energy ET). The object triggers are formed by applying thresholds to individual calorimeter towers. Electron and photon triggers are formed by applying thresholds to the electromagnetic energy of a tower. The jet triggers are formed by applying the thresholds to the electromagnetic and hadronic energies of the tower. The Level-1 jet triggers require a single trigger tower as 0.2 x 0.3 in (17,45) space to be above an ET threshold. There are two Level-1 triggers that feed the jet triggers: STT5 and STTlO. STT5 requires a single trigger tower above 5 GeV and STTlO requires a single trigger tower above 10 GeV. These thresholds are typically _<_ 33% of the Level-2 cluster ET requirment and thus have a negligible effect on the combined trigger efficiency. 50 Prescales and Rate Limiting In addition to imposing the aforementioned requirements to select out interesting physics events from the minimum bias events, triggers can be prescaled. The prescal- ing can be done either by simply accepting a predefined fraction of the events that satisfy the trigger or by limiting the rate at which the trigger events are recorded. The jet triggers are prescaled by accepting a fixed fraction of events. 5.2.2 The Level 2 Trigger Jets are not fully contained in a single calorimeter tower. A single tower will only contain a fraction of the total jet energy. The Level-1 thresholds must be set lower than the jet energy to provide an efficient trigger. To avoid saturating the Level-2 bandwidth while spanning a wide range of ET, four jet trigger samples were collected using Level-2 cluster thresholds of 15,40,60 and 90 GeV and nominal prescale factors of 240,50,20 and 1. These are used to form the jet trigger samples, Jet20, Jet50, Jet70 and Jet100 respectively. The clusters are found by the Level-2 cluster finder. In this algorithm contiguous regions of calorimeter towers with non trivial energy are clustered together. Each cluster starts with a tower above a seed threshold and all towers above a second, somewhat lower threshold that form a contiguous region with the seed tower are added to the cluster. The size of the cluster expands until no towers adjacent to the cluster have energy over the second threshold. Once the entire cluster is found, the tower energies are removed from the list and the next seed tower found and the algorithm is repeated. For each cluster found the total EM and HAD energies are calculated and recorded with the number of towers and the (17, (b) of the seed tower. The data from the calorimeter are collected and processed by the Level-1 trigger. 51 The towers are summed on the detector into trigger towers of 0.2 x 15 deg in (77, p). This is a 24 x 24 array (1152 towers , 576 EM, 576 HAD ). The tower energies are weighted by sin 0 and are gain and offset corrected. 5.2.3 The Level 3 Trigger CDF uses the JetClu algorithm in the Level-3 jet trigger software. Here we outline the implementation of the algorithm in the trigger. The CDF calorimeter has 84 pseudo-rapidity (77) annuli covering the 77 range -—4.2 2 77 2 4.2. These 77 annuli are divided into 24 azimuthal (<75) segments in the [77] 2 1.2 regions and 72 segments in the [77] g 1.2 regions. The JetClu algorithm consists of 4 stages: preclustering, clustering, splitting/ merging and the calculation of jet parameters. This is done using Z = 0.0. L1 Trigger L2 Trigger L3 Trigger ST5 (20) CL15 (12, 25) J20 CL40 (1) J50 ST10 (1) CL60 (8) J70 CL90 (1) J 100 Table 5.1: The trigger paths and prescales { given in parenthesis ) used in the analysis. 5.3 Trigger Efficiency The efficiency of the jet triggers is dominated by the Level-2 triggers. The Level-2 clustering algorithm is a nearest neighbour clustering algorithm. The Level-2 cluster- ing and the JetClu algorithm which is used in the Level-3 trigger are quite different. We now further complicate things by using the Midpoint algorithm which differs from both the Level-2 clustering and the Level-3 JetClu clustering. For each of the jet triggers the efficiency of the Level-2 cluster ET cut is measured 52 as a function of the Midpoint algorithm jet PT. The overlap of separate trigger sam- ples allows the derivation of the trigger efficiency curves. As an example consider the Jet50 trigger efficiency. The efficiency curve (cm-9(PT)) for this trigger can be found by dividing the PT distribution of Jet20 events that satisfy the Level-2 requirement (19977“-2 C’uste" > 40GeV ) by the full Jet20 PT distribution: Ctrig = (5'1) where Mm-g is the number of events in the subsample that passed the Level-2 cluster ET requirement. Mtotal is the total number of events in the parent sample. The uncertainty in a given trigger efficiency bin is calculated binomially: 6(1 — e) as = —. (5.2 Mtotal _ 1' ) The efficiency of the Level-3 ET cut is found in the same way as the efficiency of the Level-2 cut. The Level-3 ET requirements are not very different from the Level-2 cluster requirement, thus the difference between Level-2 and Level-3 trigger efficiencies is not very large. The trigger efficiency curves are calculated and fitted to the function 1 C(PT) = 1 + e(-P1(PT+P2))' (5.3) We select the PT regions of the jet trigger samples that satisfy etrig(Level —3) > 0.995 using equation 5.3. 53 — Level-2 Eff. — --- Level-3 Eff. 0. 2030405060708090100 P. (GeV) M70 1'02 I'UIIIU'IIIUIIITUITIIIIIYIIIIIIII'IITITd 1 JV 0 Q I --------------------- :a---------q 0.98 ‘ - 50.98 - — Level-2 Eff. -‘ t - . f— . 0.94 P - - - Level-3 Eff. 00 00 100120140 160100200220240 p,(cm 0.92 IIIIIITTTIUTIIrfTIIIIII M11 — Level-2 Eff. l l l l l --- Level-3 Eff. 100 120 140 PT (GeV) 406000 M100 1 .0" Ull'llIIIIUIVIT‘YIU'VI'UIIIIVIIIIITIIIU .- j J s' l 00-96 - L' — Level-2 Eff. .. I f 3 0.94% - - ---LeveI-3 Eff. . 0.92!— E _ 0.9;111111111I’nnlnnllnnllnnnlnlllllllnlLLl‘ 60 ”100120140100180200220240 P.(GeV) Figure 5.1: Trigger efficiencies for the jet samples used in the analysis. Both the Level-2 and Level-3 efficiencies are shown. The Jet20 efficiency was measured from the STT5 sample. The 99% and 98% efficiency values for Level-2 and Level-3 ET cuts are listed in table 5.2. 54 Sample Jet20 Jet50 Jet70 Jet100 L3 99% Eff (GeV) L3 98% Eff (GeV) L2 99% Eff (GeV) L2 98% Eff (GeV) 47.04 42.69 46.20 42.02 72.67 67.46 69.48 64.59 93.59 86.17 88.67 82.04 124.52 118.25 121.67 113.86 PreScale 413 20 1 Table 5.2: Trigger efliciencies and prescales for jet triggers. The analysis requires the Level 3 efficiency to be > 0.99. Once we know the PT value corresponding to that efficiency we begin using the trigger sample in the range PT(€ > 0.99) +5% to account for the jet energy scale uncertainty. 55 5.4 Prescales The Jet triggers are prescaled at Level 1 and Level 2 (see 5.1). We measure the combined Level 1 and Level 2 prescales from the data. This allows us to combine and use the four Jet samples to construct the inclusive jet cross section as a function of PT. The effective prescale is determined for each of the low ET trigger samples by normalisation to the next highest ET sample in the PT range where both samples have full trigger efficiency. It is important that these datasets do not contain any bad runs, especially so in the lower ET trigger. Any had run may lead to a non statistical effect in the high PT region where the number of events may be small. If this happens it may become difficult to distinguish statistical effects from systematic problems. At lower PT this will be less important as the physics cross section is large enough that. a few bad runs will not make an appreciable difference. The prescales for Jet50, Jet70 and Jet100 are consistent with the nominal values. The Jet20 prescale was changed at run 153067 so an effective prescale which depends on the data sample is used. 56 D V U V 10:I II I l I IerTITrIrrTI I II I I T T I I I I 1 I I T 9 5L 2‘ " ' 9: 8 22 l 0 8 5 .. 93 ] § 8.55 0 0 g 2 J] L‘ '3 3- F [ F : g 1 + 0 .] €75; [i ‘1: "00 ]’ h 1; 7E- ” g 16 { 0 E b %6.5'E' ] g 14 6:- — Nominal Prescale 8.0. o' — Nominal Prescale = 20.0 5.53 12 I 5: 1 1111111 ilnllanLrhh Lt1oz 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 1 1.5 2 2.5 3 P. (GeV) P. (GeV) c) - 5X10: , ‘ I I I I I I ITT—I l—Ij I l I I I rIj I: o 3 A45 . -j b -_A 20 x 0(Jetfgiyo(det20 b. 3 i 2 5 —— Effective Prescale 413i 4 _- 2 :R102 0.8 ‘I 1.2 1.4 1.6 1.8 2 PT (GeV) Figure 5.2: Normalised prescale factor for Jet20, Jet50, Jet70 and Jet100 triggers as a function of PT. The nominal prescale factors are included prior to the fit. These prescale factors are used to correct the cross section in such a way that all the triggers are normalised to the Jet100 trigger. Deviations of the ratio from unity can arise from missing data and in the case of the Jet20 sample the L2 prescale was changed during the run period. 57 1oIIIIIIIII]ITII]IIII]IIII]IIIIll 10 - —— Jet100 8 i. [ L = 218 pb". 10 ‘ :3 ----- Jet70 After Prescale 7 Q2; 10 . ---------- Jet50 After Prescale 8 E_ ,:L 10 ----- Jet20 After Prescale s 10 +3 4 '3 10 Z s 10 -'_ .-_ _ I 102 5180.619“ GeV ;,:" r: 10 MetSlg<6 GeV"2 :53 1 Il ' O _II-I-I-I-I-I-I-ICI-I-I-I-I-I-I-I-I II.‘ ..r-gtCouooouo-cccaououllI... I... IOIIIICDIU C- CD- 1 - |Z|<60 cm 10-1 _ 0.1<|Y|<0.7 0.0.-colon. --------—--- .2Llllllllllljllifillfl ILIII 10 0 100 200 300 400 1500 600 PT (GeV) Figure 5.3: Inclusive jet PT spectrum for Jet20, Jet50, Jet70 and Jet100 after applying prescales. All triggers are normalised to the Jet100 trigger which has no prescale. 58 Chapter 6 Multiple Interactions 85 Underlying Event 6.1 Introduction The inclusive jet analysis uses the Midpoint jet clustering algorithm for jet recon- struction. In each event, spectator interactions can contribute energy to the jet cone. This energy must be subtracted in orderto compare to LO and NLO perturbative QCD models that do not contain multiple interactions and/or underlying event. In the following chapter we present the measurement of the multiple interaction and un- derlying event contribution to jets. The multiple interaction correction derived from the data is used to correct the jets; however, the underlying event correction we use is taken from Monte Carlo. The underlying event study from the data, including the PT(:l:90°) cone study is included for completeness only. Most of the inelastic pp collisions from the Tevatron result in soft collisions with most of the particles going in a direction only slightly deviated from the original p or p directions. Such events are triggered on using the CLC (see section 4.3.5) and make up the minimum bias sample. In addition to these events there are hard scattering events which produce jets. The underlying event is everything except the hard scattered jets, it contains beam 59 remnant plus contributions from initial and final state radiation. In addition to the soft spectator interactions there can be multiple hard interactions associated with a single bunch crossing. 6.2 Data Set In this measurement of the multiple interaction correction and the underlying event momentum we use the tower PT and jet PT which has been corrected using the primary vertex as the interaction point. The event properties of the minimum bias sample are shown in figures 6.1 and 6.2. The events are required to pass the following cuts: ‘ [Zvertexl < 60 cm 1 0 Missing ET < 30 GeV. The data span the same running period as the data sample used in the inclusive jet analysis. This ensures the measurment samples the same or at least similar instanta- neous luminosity as the jet samples that it is used to correct. 6.3 Method 6.3.1 Minimum Bias Momentum in a Random Cone The multiple interaction correction is measured in the data and applied to the raw jet energy. The absolute corrections are derived from the Monte Carlo, as are the underlying event corrections as well. The application of these later two corrections are discussed in chapter 9. 60 The multiple interaction correction is measured by considering the minimum bias momentum in a cone placed randomly in (Y, (b) with the constraint that 0.1 < [Y] < 0.7. The cone energy is measured as a function of the number of quality 12 vertices (NQ12Zv) in the event. Quality 12 vertices are required to have 2 2 COT tracks. The slope, A1, of the straight line fit to (Pflame versus number of NQ12Zv is the energy that needs to be removed from the raw jet PT when more than one vertex is seen in a jet event. The correction has the form: UEM(R) = Al x (NQlZZv - 1). (6.1) 6.3.2 Summing of Tower Momentum in a Cone The towers that are within the cone must be summed in a manner consistent with the Midpoint jet algorithm clustering. The summation of towers uses the following pre- scription: for each tower construct the 4-vectors for the hadronic and electromagnetic compartments: Ptower : (P17: Py1P27E)HGd + (Pm’ Py’ Pz’ E)EM 1 Etower +p§0wer vim? = §zn(E,mr_ PM”) (6.2) where: Pz(had/em) = Ehad/em3m(9had/em)008(<13) Py(had/em) = Ehad/emsin(6had/em)37:”(95) P2(had/8m) = Ehad/em008(9had/em) (1%)2 = (gem—(210.)? (6.3) 61 where Oth/em are the angles calculated with the correct Z vertex. The azimuthal angle <73 is the same for the hadronic and electromagnetic tower compartments. 6.3.3 Effect of Single Tower Threshold In the Midpoint algorithm all towers with a PT > 100 MeV, within a radius R = 0.7 of the jet centroid, are included in the jet momentum. The non-jet momentum can contribute to this momentum in two ways: a non-jet tower above the threshold is just added to the jet momentum or a non-jet tower below the threshold can contribute if it overlaps with a jet tower and the combined PT is above threshold. Instead of doing a full study by mixing jet events with minimum bias events, we compare the momentum in the R = 0.7 cone using three different single tower thresholds; 50 MeV, 100 MeV and 150 MeV. The mean number of towers in a midpoint jet is approximately 20 with a maximum mg 50. We see in this study that the mean number of towers (P11: 0w" > 50 MeV) in a R = 0.7 cone in a minbias event is ~ 3.86, this tower occupancy decreases to 2.78 towers for a 100 MeV tower threshold and to 2.05 for a 150 MeV tower threshold. The corresponding change in the average PT found in the cones is 0.96 GeV for the 50 MeV threshold and dr0ps to 0.88 GeV and then to 0.80 GeV for the 100 MeV and 150 MeV thresholds respectively. Note that these numbers have no requirement on the number of Z vertices seen in the event. 6.4 Multiple Interaction and Underlying Event Sub- traction In the context of the full jet correction machinery, the corrected jet energy is given as: mem) = (Pfawm) >< frel(R) - U EM (13)) X fabs - U E (R) + 00(3), (6-4) 62 where Pf“w(R) is the raw PT of the jet. The relative correction (fret) is used to map the 0.7 < [77] < 0.1 calorimeter response to the response of the central calorimeter. In the current analysis, only the central calorimetery is used so the relative correc- tions are not required. The absolute correction (fabs) corrects for the energy from a hadron that is not sampled in the detector. The multiple interaction correction (U EM ) removes contributions to the jet energy due to additional hard interactions in a single crossing. The underlying event correction U E removes the contribution to the jet energy from soft spectator interactions and beam remnants. The out of cone correction (OC(R)) corrects for energy lost out of the clustering cone by adding energy back in. All corrections depend on cone size (R). 6.5 Results Here we sumarise the measurement of the underlying event energy and the multiple interaction energy. The correction that is applied in the inclusive analysis is based on the 100 MeV tower threshold measurement. N42122:; (PT) (GeV)(50MeV) (PT) (GeV)(100MeV) (PT) (GeV)(150MeV) 0 0.268424 4 0.0002 0.23546 40.00021 0.201907 40.00020 1 1.01558 4 0.00038 0.93705 40.00037 0.847977 40.00036 2 2.00251 4 0.00140 1.86343 40.0013 1.70014 40.0013 3 2.99476 4 0.00512 2.80485 40.0050 2.57705 40.0049 4 3.97096 4 0.01722 3.7401 40.016 3.45628 40.016 5 4.96319 4 0.04768 4.70266 40.047 4.37765 40.046 6 5.61756 4 0.10616 5.33413 40.105 4.9803 40.10 7 6.56919 4 0.23632 6.27037 40.234 5.87383 40.23 8 7.90464 4 0.49985 7.57915 40.497 7.15621 40.49 Slope 0.987 4 0.001 0.928 4 0.001 0.855 4 0.001 Table 6.1: The N Q12Zv = 1 row is a measure of the underlying event for the three tower thresholds. The slope of the linear fit to (PT) versus NQ12Zv is the multiple interaction correction. 63 44.4,....,4...,....,....,..T. s 10 Entr1686191711 5 Mean 14.42 10 4 EMS 15.25 0 3 3 10 z 2 10 10 1.. . . ...Alan 0 50 100 150 200 250 300 2PT(GeV) b) ....,.....444.,r4r.,....,.... 5 EntriesG1 9171 1 10_ _ : Mean 4.014 4 . 10 ms 2.36 — 33‘ a a) 10 3 2 z 10 10 1lllllLllllLllllMllllllelLJ; 0 5 10 15 20 25 30 MET(GeV) Figure 6.1: Event properties of the minimum bias sample: a) EFT, b) Missing ET, c) leading jet PT and d) the jet multiplicity. The events are required to pass a Zvertez; missing ET and lead jet PT cut. 64 figure 6.1 continued. 6) N(Events) d) N(Events) 6 4 '1' 1 .,. ,....,....,....,....,..4...r. 1 0 Entries6191711 105 Mean 6.601 4 EMS 3.737 10 3 10 2 10 10 1 Illll ll ll 1 l 1 I III! lllll-IJ-fi-‘fljlnnj 0 10 20 30 40 50 60 70 80 90 100 Lead Jet PT (GeV) x105 25:"117 lfifi'ITr l‘ l 'I"'I"'l"'I".'__ : Entries5902903 : zoi Mean 2.462 _: 3 ms 1.867 3 15:- j: I 1 10_— _: 5:. .7 C 2 0 .1. 1 1 ..1. 1...14..1. .“ 0 2 4 6 8 10 12 14 16 18 20 "Jets 65 IIITI‘IIIIWTTTITM'IIIIIIIIIrIinTIIIII'I Entries 1.438e+07 Mean 3.862 RMS 3.766 N(Events) LlllllllLU l l l l l l l l 1 L1 .L l J 20 30 40 50 60 70 80 90 100 NTM8(>50MeV) d) Entries 1.4389+07 Mean 0.9596 RMS 1.25 N(Events) 50 60 70 80 90 1 P, (>50MeV) Figure 6.2: Effect of tower threshold on the minimum bias momentum (d-e) and number of towers (a-c) in a random cone. Increasing the single tower threshold from 50 to 100 MeV then to 150 MeV gives a decrease in (PT) in the random cone of 75 MeV and 85 MeV respectively. 66 figure 6.2 continued. b) , , . ,....,....,fi..,... Entries 1.438e+07 43 Mean 2.766 0 a HMS 2.994 2 nIIllllllllllllllllllILlJllllll 40 50 60 70 80 90 100 NTW(>100MeV) e) , 1O ‘ I l"”f""l 1"r‘l"fil""l HT 4 a 10‘. 1 (,5 Entrles 1.438e-I-07 4;: 105, Mean 0.8846 3 E: 10 __ HMS 1.21 2 2 10 10 1 llllllllllllllll ll 0 10 20 30 40 50 60 70 80 90 100 PT(>100MeV) 67 figure 6.2 continued. N(Events) N(Events) 1 JLllllllllllllll , ,...,. ,....,....,.... Entries 1.438e+07 Mean 2.052 RMS 2.416 lLlllllllllllllllllllllLllllll 0 10 20 30 4O 50 60 70 80 90 L100 Nrmn(>150MeV) 107 11.71115....111111,.1.11...., [1114,1111] 10°: 1 05 Entries 1.4366407 10‘ Mean 0.7997 1'03 EMS 1.161 102 10 1 e .11.. 1 0 10 20 30 40 50 60 70 80 90 100 P, (>150MeV) 68 a) (NG122V=0) IIII IrTI IITI IIII II—fi IIII IIII III I I I I I I I I 6 10 5 Entries9341494 10 A Mean 0.9371 1‘9 10‘ 5 3 ans 1.15 E”: 10 z 102 10 1 'I.4111I441111n11111m111.11.. 0102030405060708090100 PT(GeV) b) (N012Vz=1) 5 'FTIH'W'H'I‘WTI'T"I'H'I'"'IH'W'H'I'T' 10 -‘ 4 Entries1398552 10 — E 3 Mean 1.663 8 10 ans 1.626 > I." 2 z 10 10 1 Ill III 010 20 30 40 50 60 70 80 90100 PT(GeV) Figure 6.3: PT in random cone R = 0.7 with quality 12 vertex multiplicity require- ments ( tower threshold 100 Me V). 69 figure 6.3 continued. c) (NQ12Vz=2) 4 ""l""I""I""I"T’I""l""l""l""I"" 10 " Entries161498 3 310 Mean 2.805 H C q, 2 ms 2.026 > - _ 9.". 10 z 10 1 1| 0 10 20 30 40 506070 8090100 Entries 19291 Mean 3.74 RMS 2.36 I llllllll l I Illllll 70 figure 6.3 continued. 6) (N01 22V=4) TI—Ifll 'IIIIIIIIIIIIIIlIIIT'IITrIIIII]IIIIrIiIfilT‘ a 10 4:. Entries 19291 I A 2 Mean 3.74 '0 10 .— =- E = = o : RMS 2.36 : > _ ‘ l.l.l r- _ V z 10 '5— -§ 1 =‘ ill] ‘2 _LLLLIIII lllLllllLlLlllllllllllLLLLLllLllllMLl— 0 10 20‘304050607080901 PT(GeV) f) (N012V2=5) l I I I I I I I I I I I I I I l I I I I I I I I I I I I I I I I I I I I I I T T 2 1 0 Entries 3381 -.-: I I“? Mean 4.703 -1 E _ q, ' HMS 2.743 I > 1.1.1 10 :2— '3 v : 2' Z - .. l— — : ll“ 2 I l l I 1 L L l l l l I 14 l 1 I l L L L l l 1 LL 1 14 LA I I l l : 010 20 30 40 50 60 70 80 90100 PT(GeV) 71 figure 6.3 continued. g) (NQ12VZ=6) 2 unprnl....lnnynrtlnn,‘”Hqummlnn 1o ‘3 Entries 837 I 3 Mean 5.334 - “ § 10 HMS 3.053 ':_: Ill 2 v —1 2 fl .1 1 II I ‘5 7 ll llLLllllllllLlllllllllJlLlll [4 ll‘ 30 4O 50 60 70 80 90 100 PT(GeV) 72 8§TIIIIIIIIIIIIIIIIIIIIIII1IIIIIjTTIIIIIIIIIIIIII E UE(100) = 0.94 GeV 2 7 :- UEM(100)= 0.93 GeV 0 6 E UE(150) = 0.05 GeV $ 3 E" UEM(150) = 0.86 GeV 1: 5 E_ UE(50)= 1.02 GeV - 9 E UEM(50) = 0.99 GeV .5 4 :— 2 E 0.1<|v°°"'|<0.7 10. '- 3 :- V : A 150 N 2 L. ---: Fit to tower threshold 150 MeV v : e 100 Z — Fit to tower threshold 100 MeV 1.— o 50 I --- Fit to tower threshold 50 MeV om]!IllllllLlilJllllllllllllllllljllulllllllll 0 1 2 3 4 5 6 7 8 9 NQ122V Figure 6.4: Linear fit to minimum bias momentum versus number of quality 12 vertices for tower thresholds of 50, 100 and 150 MeV. 73 8 ii I I I I T I I I I I l TfI I1 I I I I I I I I I l I I T I I I I I I I I I I I I I I I I t 5 UE(Pt) = 0.94 GeV —*— g 0: 7 7 0151mm): 0.99 GeV -: S E 1?: _; 0 5 :- . _ ‘2 1: : : 5 5 :_ UE(Et) = 0.97 GeV _: h E UEM(Et) = 0.98 GeV 3 .E 4 E. -: 2 1- 3 E 0.1<|v°°"°|<0.7 g in. r —_ V : 0 ET Rand Cone R=0.7 -_~ N 2:— --.Fitto(ET) :5 V E v PT Rand cone R=0.7: P$=(E Px)2+(2 P923 1 _— _Fit to( PT) —.: I;T?7Llljllllll llLlllLll IllllllLJ U+1llllllLl llll; 0 0 1 2 3 4 5 6 7 8 9 NQ12ZV Figure 6.5: Linear fit to minimum bias momentum and energy versus number of quality 12 vertices for tower threshold of 100 Me V. 74 6.5.1 Corrections Versus Instantaneous Luminosity In this section we recalculate the UE and UEM corrections in 6 bins of instantaneous luminosity and check that the number of observed vertices is directly proportional to the instantaneous luminosity. The proportionality of the number of quality 12 vertices to the instantaneous luminosity is consistent with the assumption that these vertices characterise the number of interactions observed in a given event. This is true when the number of vertices in the event is less than 4—5; above this there may be a problem with fake rate. The jet samples are dominated by events with g 2 vertices so any non-linearity seen with a large number of vertices in an event should not effect the cross section correction. We also observe that there is only a very weak dependence of the correction factors on the instantaneous luminosity. 75 4°. q .1 _J ‘ ..Ir...,rfi.,....,.n.,fl..,....,....,... 90 80 70 60 50 40 30 20 10 o lllllllllLLJllllllll lllllllllllLJilllJllllJl 010 20 30 40 50 60 70 8090100 Inst Lum x 103° cm’2.s1 NEvent Ill IlllIlllllllllllnlllllllllllllllllllllIll i llllllllIll]lllllllllllllllllllllllllllllll‘ llll U' v 'I'rllllrttllrrv - — q _ q — d .1 fl .1 d .1 — d _ llllll‘ Inst Lum x 103° cm"".s1 l:lllllllllllllllJllLllllllllllllllll P _ P 1. 1.. l- b 1- l. _ _ P — _ O .s N a h 0'. fi ‘1 0 ID z D _L N < Figure 6.6: a) Instantaneous luminosity for the min bias sample for the same running period as the jet triggers used in the cross section analysis. b) Instantaneous lumi- nosity as a function of number of quality 12 vertices. c) Average number of towers in a random cone R = 0.7 as a function of the instantaneous luminosity and d) average PT in a random cone R = 0.7 as a function of the instantaneous luminosity. 76 figure 6.6 continued. C) ( "Towers ) (PT) - -1 — _ d _- d — .— d d q I - d - - u .— I I I l _I‘ 4.5 5- ll J ... :1 4 : ”fit : sewn" ; 3.5 F fed-”P —E : _P_,..- I: 3 E- ____,.- 'E L— -‘ — 2.5 E- __—I'-d- i E” g 2 _— __ 1.5 E- —Z 15— é 0.5 E- -E r— -I o I I 141 I I I l I l l I I I I I I I I I I I I I i I I I I I I I I I L I I I I I I I I I I I I I F 010 20 3040 5060 70 8090 Inst Lum x 103° cm".s1 d .I —II q ‘ d — q — d a: q CII q —l .1 ..l .4 d .1 — q .1 —I d j 1.4 1 .2 ”Nah 0.8 _.-"' 0.6 0.4 0.2 oIIIIlIIIIlIIIIIIIIJIIIIIlIIIIlIIIIlI IIlII , I L I l I I 0 10 20 30 40 50 60 70 80 90 Inst Lum x 103° ¢:m'2.s1 'lllllllllllllllllllllljllllll 77 0 100 Me V) as a function of number of quality 12 vertices. Fit is to p0 +171 X N Q12Vz. The slope gives the multiple interation correction. The fits are done in 6' bins of instantaneous luminosity, see table for details. 78 Figure 6.7 continued. 1 530 x 1030 cm-2.s1 9 I IIIIIIII—rrIIII'II IIIIIIIIIIIIIT—IIIII'IIIIIII I p0 0.05426 1 0.00387 + p1 0.9388 :1: 0.003099 + - IIIIIIIIF (PT ) in 13:07 IIIHIIIHIIlIllllllIIllllqllllllI IIIIILIIIIIIJIIIIIIILL .1...2....3. 4””5 6 7 8 9 Number of 012 Zv odnuhmmNm I O} .- l 80 1 .M E: I I I I I I I III I r f f r I I I I I I I I I I I I I I I I I I I _E 1'02 :— - UE (Pl°“'°3100 MeV) '3 3 1 E7 o UEM (PI°"°'>100 MeV) _._i (D 3'- 1 v0.98 —- — f: : Last Bin —> Lum 2 30x 103° cm".s‘ : d 0.96 L— _'.‘ .11 _ _.__ 1 I“x084 :— ——¢—-§ : T- ‘ __.__ __¢_ : £0.92 9% : _¢__ -: m I ”—9— j :3 0.9 _-_— 1 0.88; if 0.86 :— -: — I I I J l I I I I J I L LI L I I I _I l I L I I l I I I I I I I I I d 0 5 10 15 20 25 30 35 Inst Lum x 10’“ ¢:m'2.s1 Figure 6.8: Multiple interaction correction and underlying event as a function of the instantaneous luminosity. We see a weak dependance on the luminosity. The effect of the luminosity dependence is accounted for in the systematic assigned to the multiple interaction correction (see chapter 12). 81 6.5.2 Instantaneous Luminosity & Number of Quality 12 Ver- tices in Jet Samples .161100 35;19’..m,...., ,fi . mu: 30 LE— Entries 2375921 3 E nus 8.551 E 5 20: -: > - _. I.|.l ' ‘1 z 15:. ; t : 1— —l 10:— -E 5E- -E %—I IlIIIIlLILIlIIIIIIIIJIIJIIIIIIIIIIIIJIIII _ 5 10 15 20 25 30 35 40 45 50 Inst Lum (x 103° cm'2.s“) Jet70 x103 '1 1'“ 1 1 'I'r‘rin'wnnl'n‘rfl'_ 25 Entries 1913696 .2 Mean 20.89 E 20 -—_- .5 11115 8.475 : .i.’ 15 a z 1 10 —j 5 1 % IIIIIII l InllenLLilnInjlnnnnlrlJn . 5 10 15 20 25 30 35 40 45 50 Inst Lum (x 103° cm'2.s") Figure 6.9: Instantaneous Luminosity of the jet trigger samples. These distributions are consistent with the corresponding minimum bias distribution that was used to determine the multiple interaction correction. 82 Figure 6.9 continued. Jet50 x103 605‘ Entries4516388 50:- Mean 21.15 .5 405— 11115 8.569 0 _. a, 1: z 30*: 205— 10}- :1 IIIIIII_L IILIIIIIILIIIIII ILJIIIIIIIII 00 51015 20 25 30 35 4045 50 Inst Lum (x 103° cm'2.s") Jet20 x10‘ 16:— 14:— 12:— 5 10L 3 I z'" 35‘ GE- 4:— 2:— 00‘ 5 10 15 20'”25 30 35 40 45 50 Inst Lum (x 103° cm‘2.s“) 83 Jet1 00 x104 1on_..4,....r.....,...,...,.......,...,..._ 80:- Entries1946649 _j - Mean 1.817 3 § 60:" —7 RMS 1.145 ‘3 2.. 40; _1 r _ — -l 01 .1...1... 9.1..4L...1...1...1...‘ 0 2 4 6 8 10 12 14 16 18 20 N012Zv Jet70 4 xI1II'IIIIIrijIIIIIIIIIIIIIIIIIjIrIIIITI 90:— -= 80E 5 '5" Entries1701536 ‘5 70:— -I 5 Mean 1.778 E 2' 60'5— E g 50;— L_ HMS 1.146 _g i" 40:- -: ..;_ -: 20E— 4} 10E- a: o. .in..1..L -nm.1...1...1...1.1. m3 0 2 4 6 8 10 12 14 16 18 20 NQ122V Figure 6.10: Number of vertices per event in the jet samples and minbias sample. The jet samples are dominated by NQIZZv of 1 and 2. So any non-linearity of the correction correction quantity ((PT) versus NQIZZv) should not affect the corrected cross section. 84 figure 6.10 continued. Jet50 5 x10 I I I I I 22 _— 20 18 16 14 12 "Event Entries 4113290 Mean 1.769 RMS 1.178 IIIIIIJIIJIIIIIIIIIllllllllllllllllllllllllll I IIL IlIL nut con-ham: Jet20 . cm .1 - ...I.. 1 1114.111; 468101214161820 N0122v 60 50 40 I 1 —-1 .1 ...1 q fl .4 I —1 —1 -1 -—1 .1 30 NEVOI‘II 20 10 IIIIIIIIIllllllllllllIIllTIIIlll 5 Entries 1.1244020-407 Moan 1.723 HMS 1.182 llllllllllllllllllllllllllllJlIllq I l I I 1 L4 I O I .1. ...l.. IIIIIIIII 4 6 8 10 12 14 16 18 20 N0122v 85 figure 6.10 continued. MinBias 45 40 35 30 25 20 15 10 5 o 1 1 1 0| —1 I .1 _ - q d d — d d .1 ..1 .1 - .1 .1 .1 Entries 5902903 Mean 1.21 RMS 0.5837 NEvent [IfIIHHIIHIIIIIIIIIIIFTTIIHIIIIITIIHI 5 lllllIILIILIIIIIlllllllllLlJllJlllllllllIll ...1..41...1...1...1..L-1 10 12 14 16 18 20 NQ12ZV O M b O m 86 6.6 90° Transverse Momentum (P190) in Jet Events For each event, the transverse momentum and the number of towers in a cone R = 0.7 centered at n = 173°”, 05 = (#71211 :1: 90° is calculated. We use the same two tower thresholds as in the previous sections (50 and 100 MeV). Here we still use the minimum bias sample and select a ”jet sample” from it by requiring that an event has a central jet with PTJet > 5 GeV. This sub-sample of the min bias data is used to study the P790 in the low energy jet events. The 90° cones includes energy from (a) jet activity, (b) energy from soft interactions from spectator partons and (c) additional interactions occuring in the same bunch crossing. In order to isolate the contribution from the jet activity we consider the 05 + 90° and the d) — 90° cones. The difference of these cones is related to the jet activity [33]. We wish to compare the P520 of the ”min-cone” with the underlying event found from the full minimum bias sample using the random cone study. 87 a) 100MeV 105 : 5 TIIIIIIIIIIIITIIIIIllIII—IITIIIIIIIrIIIIIIIIIIIjI—I Entries 702740 Mean 5.339 RMS 3.024 IIIIIIlIIIIlIIIIIIIIIIIIIIIIIII ‘40 50 60 70 80 90 100 NTowers Figure 6.11: P5940 and number of towers for 50-100 MeV tower thresholds 88 figure 6.11 continued. 0) 50 MeV Entries 702740 Mean 1 .876 Entries 702740 Mean 7.22 RMS 3.739 IIIIIIIIIIIIIIIIIIIIIIIIII 0”“10"”20"'30”'40”'50 60 70 80 90 100 NTowers 89 105 IIVIIIIIlTrIIlTlti[itirrfilIIrT1ITTIIIIIIIIII 10‘ -. Entries 702740 3 Mean 1.743 310 § HMS 1.475 111 2 Z 10 10 1 ....L....1....1.” Ii! 7 0 5 10 15 20 25 30 35 40 45 50 PT¢+90)(GeV) b) ...,....,....l..r.I.+. Entries 702740 Mean 1.745 5 EMS 1.477 5 Z .111...“....1....1....1.... 30 35 40 45 50 1511-90) (GeV) Figure 6.12: P12°(¢+ 90), P720011 - 90), P520 (max), P199 (min), Pf?!) (ave) and Pg) (diff) for Pgm" > 100 MeV 90 Figure 6.12 continued. 0) 5 1O 1""IHT'IH"1'"'1""1""1'H'1'H'Pr' 1' 4 10 Entries702740 3 Mean 2.355 310 5 5 EMS 1.607 E. 2 z 10 10 1 1 ...............IUIi 11111 1.. 0 510 15 20 25 30 35 40 45 50 P$°(max)(GeV) d) I 1'1"”r""1""1fl"1' 1""1""1""1"" Entries702740 Mean 1.132 s 3 EMS 1.014 .0 z IIIlIIIflIIIIIILIIIIIIIILILLLLI[III 05101520253035404550 P‘T’°(min) (GeV) 91 Figure 6.12 continued. 6) Entries 702740 Mean 1 .744 HMS 1.212 NEvents IIIIIIIIIIIIIIILIIIJ '30 35 40 45 50 P$°(ave) (GeV) Entries 702740 Mean 1.223 RMS 1.159 NEvents P$°(max)-P$°(min) (GeV) 92 1o 'rIl""l""l""I'Ivvlvvvrrtttvlvrxyl If 10‘ Entries 702740 3 Mean 1.875 310 5 RMS 1.512 0’1 2 z 10 10 1 I“! 11111 | 05101520253035404550 P1(¢+90) (GeV) Entries 702740 Mean 1 .877 HMS 1.514 Events 20 25 30 35 40 45 50 P7011-90) (GeV) Figure 6.13: P19~°(¢+ 90), 1319.0(8 — 90), P%0(max), P190(mm), P19~O(ave) and P19.°(di f f) for PTTW’" > 50 MeV 93 Figure 6.13 continued. C) .-.,....,.-..I....,....,....,...,]-mp..... .. 4 10 Entries702740 3 Mean 2.495 3 E, nus 1.642 E 2 _ z'" 10 ‘3 10 1 llllllllLllllllllllll o 5 1015 2025 30 35 4o 45 50 P$°(max)(GeV) d) Entries702740 Mean 1.256 g RMS 1.055 > Ill 2 0””5 ‘10”‘15 20 25 so 35 4o 45 50 P$°(m1n)(GeV) 94 Figure 6.13 continued. 9) NEvents NEvents Entries 702740 Mean 1 .876 RMS 1.251 1J111u|l 1 1111 lllULlllLll 1W111Ln11l1111l1111l114112111 51015 20 25 30 35 40 45 50 P.°,°(ave) (GeV) Entries 702740 Mean 1.241 RMS 1.168 1".11.“..11..111111111114lnnn. 20 25 30 35 4o 45 so P?“(max)cP$°(min) (GeV) 95 I T I— I I I 1 I I I I l I I I I I I I - P$°(max) . P9°(min) o P$°(max)-P.9r° (mm) + + *+++++ 1 + ”e ++ H V¢ O” 1.- ...O 1111’ 11 ,’iflg;;;;zzill‘3¢$ 911$ it“ 1 + 4 1.- Min Bias PT (GeV) .15 N (A) b 0| O ‘1 Q IIIllImllIlIIllIllIII]]TIIIIIIIIITITTIIII o 1 1 l 1 l 1 1 1 1 L 14 1 1 L 1 1 1 1 o 10 20 30 4o 50 PT(Lead Jet) (GeV) Figure 6.14: P120(max), 193.0(m1'n) and P1910(dz‘ff) for Pgwer > 100 MeV as function of lead jet PT 96 IlIIIIlIIII'IjTjIITTI - P$°(max) . P$°(min) o P$°(max)-P$°(min) + ”e*+++++++++++++1 .- '0- .0- YA fir W” 11111 . “...,... wtiiifitwg + + 4 .140 ' 6": 330000" :22“ + .. Min Bias PT (GeV) .1 N (A) «5 GI 01 N G IIIIIIITIIIIIIIIIIII[ITIFITIIIIIIIIIIIIIII l L 1 J l l 1 J I l I l l l l l 1 1o 20 30 4bL“ 150 PT(Lead Jet) (GeV) OD , Figure 6.15: P7910(ma:1:), P7910(min) and Pi§0(di f f ) for Pgowe" > 50 MeV as function of lead jet PT 97 6.7 Conclusion We have studied the energy/ momentum deposition in a R = 0.7 cone in the minimum bias data and compared it to the 90° PT in the subset of minimum bias events that satisfy leet > 5.0 GeV. It was found for a cone R = 0.7: 0 On average the underlying event contribution to a jet in the central region is 0.9375 GeV. The value is measured from minimum bias data having |Z| < 60.0 cm with a tower threshold of 100 MeV, using events with a single quality 12 vertex. 0 The multiple interaction correction was found to be 0.928 GeV/ Vertex for a 100 MeV tower threshold. 0 Decreasing the tower thresholdfrom 100 to 50 MeV increases P1132”7 by ~ 100.0 MeV. o The minimum P510 in the jet events is weakly dependent on the lead jet PT in the event. The (P%0(min)) of the minbias events that have a 5.0 GeV jet is 1.13 GeV (average over full run range). 0 The maximum P5910 in the jet events increases slowly with the lead jet PT over the range included in this study. The P1910(mz'n) and P19~0(max) - P%0(min) both have weak dependence on the lead jet PT. 98 Chapter 7 Jet Energy Resolution 7 .1 Introduction To measure the jet energy resolution we use the technique of PT balance, first intro- duced by UA2 [19]. For a 2 -—> 2 event in a perfect calorimeter, momentum conserva- tion requires the PT of the first jet to be equal to the PT of the second. Calorimeter resolution and QCD radiation produce fluctuations in PT which can result in a PT imbalance for an event. This imbalance is related to the single jet resolution and it is what we measure. A vector RT is defined for the dijet system as the vector sum of the transverse momenta of the two leading jets in the event. In the absence of initial state radia- tion, conservation of momentum requires the total transverse momentum of the event (hard-scattered jets ) to be conserved. In a pure dijet event this implies that RT = 0. Detector resolution and QCD radiation can produce a momentum imbalance in the event causing KT to deviate from zero. The coordinate system for the dijet I? T is defined so that the perpendicular direc- tion (.l.) is the direction that bisects the azimuthal angle between the two jets. The parallel direction ([1) is orthogonal to (.1): I] x .l. = 2, where z is the positive 2 axis in the detector coordinate system (beam line). From these definitions we have: 99 KT“ = (P11 — P12) 1.111%), (7.11 Km = (PT1 + PT2)COS(%Z)1 (7-2) where 4512 is the angle between the two jets. The width of the KT” distribution (0”) and the KT; distribution (0;) are related to the jet resolution: the perpendicular component is dominated by QCD radiation effects and the parallel component is a combination of radiation effects and detector resolution. The detector resolution can be extracted from 01. and or“ by: URMS = T (7.3) 100 KT(Para) Figure 7.1: PT balance technique. The PT vectors of the two leading central jets in the transverse plane are shown. The _L axis is the perpendicular bisector of the angle between the two jets {(1512). The H axis is orthogonal to the 1 axis in the transverse plane. KT, the vector sum of PTI and PT2 is shown with its components along the _L and II axis. 101 7 .2 Method We measure the K T“ and K T _L distributions in the data and Pythia. The data-Pythia comparison of 03 M 5 gives an indication of how well Pythia+CDFSIM is modelling the resolution and QCD radiation. The discrepancy between data and Pythia can be used to set the size of the resolution systematic uncertainty on the cross section. Dijet events are selected so that at least one of the jets lies within the rapidity range 0.1 < Y < 0.7. This jet is referred to as the ”trigger” jet The remaining jet is referred to as the ”probe” jet. When considering the central calorimeter resolution it is also required that the probe jet satisfies the Y cut. The distributions KT” and KT .1. are constructed for central-central jet pairs. The Monte Carlo is divided into subsamples of the total available sample. The division is based on the PT coverage relative to the jet trigger we want to compare to. For example the Jet20 data is compared to a sample made up from Pythia PT 218,40,60 and 90 GeV subsamples. The subsamples are weighted by luminosity for the study. The data are required to pass the good run requirement, database-ntuple event count matching and the following cuts: “Etotal < 1960 GeV . ET < XX 0 [2] < 60cm 0 0.1 < |Ytrigger/p'0bel < 0.7 e A¢12 > 2.7 radians . P13.” 3"” < 0.1 x Pq’i‘vemge, where XX is 3.5, 5.0, 5.0 and 6.0 for Jet20, Jet50, Jet70 and Jet100 respectively. A further selection cut is made on the average PT of 102 the leading two jets to minimise any bias in the measurement. Jet20 Jet50 Jet70 Jet100 P14?" (GeV) 50 75.0 95.0 130.0 (Pd,je,) (GeV) 30-45 45-81 81-100 100+ Table 7.1: The P7443." cuts is used to ensure the Pythia samples and the data are away from generation and trigger thresholds respectively. 103 7 .3 Results 2 _- I I I I I I I I I I I I I I I I I I I I I I I I I T j I I I I I I I I I I I I - 1.5 T— . Fractional Difference in O'RMSI data and Pythia —: g «n : Fit to constant (PO) 3 E6?“ 1 -_— p0 0.05714 1 0.03366 1 ‘ ” j is" : - D“: I 0 Z I 1 1 1 1 1 l 1 1 I 3 £2 0 :— 1 1 1' 1 I 1 . T 1 l is __ b“ : 1 v _ .1 -0.5 _— —: C i -1 L I I I I I I I I LI I I I I I; I I I I I I L I I I_L I I I I I I I I I I I I 1 0 50 100 150 200 250 300 350 400 < P1> (GeV) Figure 7.2: Data-Pythia Resolution difference versus (P1117 ij 6t). 104 (P744111) — (P744056) (GeV) Klf’ata “1121f“ Kffi’m“ 5Kfif‘h‘“ 6040 0.247 0.021 0.242 0.032 8060 0.252 0.011 0.237 0.032 100-80 0.177 0.012 0.170 0.027 120100 0.566 0.027 0.606 0.109 140120 0.264 0.008 0.247 0.031 160140 0.173 0.007 0.158 0.021 180160 0.172 0.010 0.153 0.017 200180 0.161 0.015 0.152 0.016 250200 0.157 0.016 0.145 0.012 300250 0.135 0.035 0.135 0.010 350300 0.132 0.071 0.132 0.015 400350 0.106 0.174 0.126 0.028 Table 7.2: KT” for data and Pythia as a function of the average dijet energy. (3.141") - (P744613) (GeV) Kfata 5K9?“ Kfl‘h‘“ 6Kgithi“ 6040 0.063 0.021 0.059 0.032 8060 0.053 0.011 0.050 0.032 10080 0.049 0.012 0.042 0.027 120100 0.072 0.027 0.065 0.109 140120 0.067 0.008 0.057 0.031 160-140 0.063 0.007 0.058 0.021 180160 0.060 0.010 0.056 0.017 200180 0.057 0.015 0.049 0.016 250200 0.055 0.016 0.048 0.012 300250 0.049 0.035 0.044 0.010 350.300 0.051 0.071 0.044 0.015 400350 0.055 0.174 0.038 0.028 105 Table 7.3: KT_L for data and Pythia as a function of the average dijet energy. (19,4471) — (P144093) (GeV) 63353, 66,133}; egg?“ 65%: 4060 0.169 0.010 0.166 0.016 6080 0.174 0.005 0.163 0.016 80100 0.120 0.006 0.116 0.013 100120 0.397 0.013 0.426 0.054 120—140 0.180 0.004 0.169 0.015 140—160 0.114 0.003 0.104 0.010 160-180 0.114 0.005 0.101 0.008 180200 0.106 0.007 0.101 0.008 200—250 0.104 0.008 0.096 0.006 250300 0.089 0.017 0.090 0.005 300—350 0.086 0.035 0.088 0.007 350-400 0.064 0.087 0.084 0.014 Table 7.4: Jet energy resolution (03 M S ) for data and Pythia. Table 7.5: Jet energy resolution (URMS) fractional difference detween data and Pythia. (P1114171)— (P1134093) (GeV) (6,13% — efifi’gfiyogfi’g‘“ Err 4060 0.016 0.064 60-80 0.064 0.112 80100 0.035 0.132 100-120 0.068 0.124 120-140 0.062 0.101 140-160 0.096 0.118 160-180 0.130 0.109 180200 0.048 0.114 200250 0.079 0.108 250300 0.010 0.203 300350 0.024 0.411 350-400 0.240 1.032 106 SE. -‘ lllll Illll Illlll lLlLll llllllllll Illll KT Parall X —L ‘9. Events O _ N 01 llllllllllIllIllllllllllllllllllllllllllIIIIII lllllllllllllllllllllllllllIllIlllllllllllllll KT Parallel Figure 7.3: KT“ for data (histogram) and Pythia (points). This quantity is sensitive to both QCD radiation and detector resolution. All distributions are normalised to unity. 107 Figure 10.6 continued. Jet70 0.5 J—L °. q-l 0.4 0.3 NEvents 0.2 0.1 lllllllllllljllllllllllIL llllllllrlIllllllllllTIIlx 1 K.r Parallel X .16 '°. lllllllllllllllllllll A“- O N d’lllllllllllllllllllllllll KT Parallel 108 Jet20 025....f....,....1 .1 q q «1 d l 1 0.2 $0.15 + 0.05 LIJllIllllllllllIlJJlllJl IITIIIIIIIIII 0° '05 KT Perp Jet50 0.2 1 ' r r f. T 1 r T T 1 0.18_ 0.16 + 0.14 0.12 - 0.1 in 0.08 0.06 . 0.04 0.02 +_._ 00 5’ i '01.1‘ ‘ L -0123. i ’0.’"" 0. 0.5 KT Perp Figure 7.4: K T _L for data ( histogram ) and Pythia (points). This quantity is dominated by QCD radiation. All distributions are normalised to unity. 109 Figure 10.7 continued. Jet70 0.22 . f 0.2 + 0.1 8 0.1 6 0.14 15 0.1 2 z 0.1 0.08 0.06 0.04 0.02 00 1 FITIFITIIITIIITIIWII ll|+ld r f 110 lIJllLllIIIIllllllllJllllll 0.5 K, Perp 7 .4 Conclusion Pythia reproduces the KT” distributions found in the data reasonably well. The K T 1 distributions found in Pythia appear to be slightly narrower that those found in the data. This suggests the 3rd jet content of the data is different from Pythia. We compute the single jet resolution (03 M S) and find that the agreement between data and Pythia is good. The data is systematically higher than Pythia by ~ 6% (see figure 7 .2). This systematic difference will be propagated into the inclusive jet cross section systematics. 111 Chapter 8 The Raw Inclusive Jet Cross Section 8.1 Introduction Jet production at the Tevatron probes the highest momentum transfers currently available, this corresponds to large :16 and size scale of ~ 10"17 cm. This analysis is restricted to the central region, similar to that of previous CDF inclusive jet cross section publications. This chapter will describe the data sample, analysis cuts, jet kinematics, jet backgrounds and the raw inclusive jet cross section. 8.2 Data Sample The analysis uses data collected over the running period of Feb 2002 until Feb 2004. Over this period 275pb‘l of data was collected with the jet triggers (Jet20, Jet50, Jet70 and .16th0). The data are processed in version 5.3.3 of the CDF offline software. The Midpoint algorithm used for jet reconstruction uses a cone of R = 0.7 and f merge = 50%- 112 8.2.1 Run Selection Runs are required to pass the “good run criteria”. The good run list was composed by querying the database with the following requirements: SHIFTCREW_STATUS = 1 RUNCDNTROL_STATUS = 1 0FFLINE_STATUS = 1 RUNNUMBER >= 138815 CLC_STATUS = 1 L1T_STATUS = 1 L2T_STATUS = 1 L3T_STATUS = 1 CAL_STATUS = 1 CAL_0FFLINE 1 (COT_STATUS 1 0R COT_0FFLINE = 1). In addition, there is a check to make sure that the number of events in the ntuple is in agreement with the number of events recorded in the database for each run that is used. This insures the luminosity taken from the run summary provides a fair measure of the integrated luminosity. This is important to the analysis as the luminosity sets the overall normalisation of the inclusive cross section. 113 Sample Jet20 Jet50 Jet70 Jet70 OffLine Luminosity (pb)-1 275 275 275 275 Nevents 19009663 5824615 2621059 3089196 GoodRun Selection 14669594 5091120 2177599 2611157 DataBase Event Matching 11867876 4516388 1827810 2310033 Number of runs for analysis 1134 1204 1170 1167 Lum for analysis (pb)_1 206.826 225.83 207.301 218.513 Table 8.1: Number of events for each Jet trigger used in the analysis and how many pass the good run list (DQM version 5 )and database-stntuple event matching. 8.3 Event Selection The raw inclusive jet cross section is required to satisfy the following cleanup cuts. These are designed to remove background events coming from detector noise and cosmics. They also ensure the jets used in the analysis are contained in a region with good tracking coverage: 0 Good run selection (DQM version 5, no silicon) .Etotal < 1960 GeV 0 ET < XX 0 |Z| < 60cm 0 0.1< |Y|< 0.7 Where XX is 3.5, 5, 5, 6 for the jet20, jet50, jet70 and jet100 samples respectively. 114 Number events pass out Jet20 Jet50 Jet70 Jet70 Total 11867876 4516308 1827810 2310033 ET 11840114 4431622 1762526 2052939 |Z| < 60. cm 11036107 4087286 1625560 1892625 ETotal < 1960 GeV 11035915 4087095 1625497 1892525 Table 8.2: Number of events passing event selection cuts for the jet triggers (ETotal: ~ ET and Z vertex). The total number of events here is after requireing the good run requirements. 8.3.1 Z-Vertex Cut The protons and antiprotons are distributed in bunches which extend ~ 50 cm in the Z or beampipe direction. This means the resulting pp interactions can occur over a wide range of Z values. Putting it another way, there is a large luminous region. . The Z vertex information used for reconstructing the jets comes primarily from the COT. The distribution is approximately gaussian with o z 30 cm, centered near Z = 0.0 cm. To ensure good coverage events used in the analysis are required to satisfy the condition |Z| < 60.0 cm. The efficiency of this cut is determined from a fit to minimum bias data using beam shape parameters [26]. Here the efficiency of the cut is also measured for each of the four jet trigger samples used in the analysis. For the jet triggers, the efficiency of the cut is measured by applying the vertex out after all of the other event cuts have been applied. Then the efficiency is defined by as the number that pass all of the cuts divided by the number of events that passed all cuts before the vertex cut. The efficiency of this selection cut enters the raw cross section as a luminosity efficiency correction and is uniform for all of the jet triggers. 115 Sample Jet20 Jet50 Jet70 Jet100 % Eff of |Z| < 60 cm 93 92 92 92 ‘70 events with no vertex 0.4 0.1 0.1 0.1 Table 8.3: Efficiency of Z vertex cut in the jet triggers and fraction of events that satisfy the event selection cuts but have no vertex. The Z-vertex correction is taken from the beam fit parameters: the value was 95% [26'] 116 5 ’ 1 v r I j i r r 1 fit I I 1 T l r I 1 j I T 10 4 10 A 8 ‘g 10 .. . 2m 2 f... a...‘ 1 o '5' K 13' ". 1 0 1'1 ' 1.1 11“ Ei§ 1 I IEIEI‘: I I I I I I I I I I 14I I I I I I I L I I I I I I I I I I IIiolulII I -200 -150 -100 -50 0 50 100 150 200 zmm) Jet50 . 1oa IIIIIIIIIIIIIIIIIIIIIIIIlIrIfiIIIIIIIIII 4 10 3 ~10 K 5 ‘1' > / o “1 2 . ‘1, z 10 .1 '. '1' '., . 7" “3.. I g b I o 1 o '0‘}. €23}. n ' ~ I. g “'l I '1 ”'l 'l 0 1 i: : ' 1 I L I i. l I I I I L I l l 1 4L 1 L4 L 1 Alm 44 l 1 1 1 l 1 1 1 1 1 1 1 1 -200 -150 -100 -50 0 50 100 150 200 Figure 8.1: Z vertex of the four jet triggers before and after selection cuts. The peak at Z = 0.0 cm in the before plot is due to cosmics and is removed by the missing ET significance cut. 117 figure 8.1 continued. Jet20 so” m ) :u — :_ — .d— - _... _—____. . 1...: _ 1...: _ 1...: _ 1...: . 1.1. 2 m 11...... . _.._. _.._. 1: 1 I I . 11.1 kW .l JIM-a. I0 I 9x _.I Iflunuuu 15 z I. f kw“ 41 I f k1.” I T .s I 1 r .5“ I 1 1m 1 1 11 .1 N I I I I I T l [0 .l I. 15 I I I I I I 1 IIIIII lo I I. I T I 1 1 f 1 I f I .I 1.0 TI 1. 5 f . 1 1 1 1. 1. .. 1 1 Y 10 I 1 a 10 II In I I1- fll I ". I. 1" L T “‘1'. J ’ol I ”if L ...mfidy 1.0 . ”Nu-n 1 I Ian-I". '5 III 'I IIIIII I . 2.51.14 3...!” 1 II. I .I “an r I .EIrEFrpELEEtIEW J. E::_ ___E:_ EELLE 3 2 5 4 3 2 .5 4 0 0 0 0 0 1 0 0 0 0 0 0 1 1 1 1 1 1 5 1 1 1 1 1 t 2.3m e «:05 z J z -150 -100 -50 118 -200 8.4 Kinematics In this section, we plot the event properties that are used in or are related to the selection cuts for events and jet kinematic quantities. All of the plots show distri- butions before event cuts and after the final selection cuts. Note however, that not all entries in the plots make it into the inclusive cross section due to the matching of trigger samples. 119 a) Jet20 Inclusive Jet PT 1-21 1 1 1.11 1 -4 -3” -511 mm 1... . _.._.... . _I Am. H o ( I. J1 T L 0 6 P I I .II. I. w 1 .n 1 5 n. .. ...unnuununuum n. ..H .1 1.1.1.1553...” 1l .1. a... 1” I .n....”llllllllm 1.- 13 ..1 ”1. l 10 I .. 1 11 lo 1. 1. 1 .2 1 1 1 1r .11-... 1“ 1 10 r . 1111111111111111111111 1 ) I . IIIII II 1 1w w T I r 1 m ”I I... I p I I a m n. m 1r.. . _.._.... _ T. 0 v- 7 6 5 4 3 2 6 5 0 0 o o 0 0 o 11.: 9.. 0 o 0 1 1 1 1 1 1 1 0 0 Q 1 1 1 1 9 J no... m: «0.. 5 Y 120 Figure 8.2: Jet20 Kinematic variables. Inclusive PT , Y and 915 before cuts (dashed) and after cuts on |Z| < 60cm, 0.1 < IYI < 0.7, Etotal < 1960 GeV and ET < 3.5 (solid). figure 8.2 continued. __d_____~___-dq_dq_—qqu17 llllllll ...IIIcUIaunlaoalliuulllnuhl II uni-III. I6 1 ".1... M I ”1.1... A 15 I”. v I I "1...”. .14 I .11.... .13 f 1 .1 II. I .1 12 11. ..m..... i 11 ..u. H 4“ hump.—b_ ____L——F—_—_;b__1 m m 0 M O 0 00 0 0 0 2 8 6 4 2 1 1 «0.. c) Jet20 ¢ ¢ (Rads) 121 a) Jet20 Em“ ijfii NEvent : U a . n I I 1 ' ' 1 a -I I I I I L4 #4]. I I L I I I I 4L1 I 'I I I I J 0 500 1000 1 500 2000 2500 3000 E11... (GeV) b) Jet20 Missing ET III—[ji—rl IIIUFTITITTIfTYU'ITIIIIIUIIYTUUIIII 1o 10‘ 1 1o 3 10 2 10 10 4 "Event I I. '. a '. a '— :‘:ii a:: n = ' I ‘ \:.'Ihfl: ”HIN’O 1-“ NJ " 'W .Ir' 0 50 100 150 200 250 300 350400450500 :51 Figure 8.3: Jet20 Kinematic variables: Etotal: ET and I}; before the selection cuts (dashed) and after (solid). 122 figure 8.3 continued. c) Jet20 Mlsslng ET significance III I III [Trl I [j g m -11- z .."I- -.- --'-l- --.-‘. - -' --.- .- '.' I.--l_" g“. _ - I--:--:-.-. I I III IILLLJIIIIIIIIIIIIIILIIII 6 8 10 12 14 16 18 20 E1" 123 a) Jet50 Inclusive Jet PT 7 10 .rlfi.r,.... e 10 s 10 4 10 3 g 102 Z 10 1O - - .‘: -" .'.-"_I'. ---_.'..0.--_I‘ 1 -1 10 [m '2 .. ..1.II;LI.LI. 10 500 600 700 PT(GeV) b) Jet50 Y (Rapldity) ..HHIWW' I l I I IW'II'IT'I 10. E- '1. "__.-"-' - —: C '_ : - ‘ _rr" m - "gm 1 § ' E E ‘ z - a a - ‘05 :— E i E E _. g i 1 IJ L141....l....l .11.. l l J.lLLl..1. -5 -4 -3 -2 -1 0 1 2 3 4 5 Y Figure 8.4: Jet50 Kinematic variables: Inclusive PT , Y and a3 before cuts (dashed) and after cuts on |Z| < 60cm, 0.1 < |Y| < 0.7, Etotal < 1960 GeV and ET < 4 (solid). 124 figure 8.4 continued. c) Jet50 ¢ 4500 x 3500 3000 g 2500 Z 2000 1 500 1 000 '1 l I If W r17 I I Ifi _ - - -|.. Q - —I 3.. - u - - g - - -. .. ."-. I: ..I.--II - .' : 4000—332:-:’-::..=::'::- ._: -- --~-.:; _ : H . "-- II "..I '-' . : I— ‘ I .— l— ' _ —- . _ E -: _ '- — _ I u- L— g 1 - . - : ' 2 L.— . 1" n— I — _ ' — I— . _ t a : _ . _ C . : I— . _ _ . - ~—— . _ l— ' d —m—_. — ._ ‘ . : - I 500- _: p _ 3 .. q I I I I I I I I I I I I I I I I I IJ I l I I I LI I I I ‘- 125 e A I D Q (0 v a) Jet50 Emu "Event to o; H'soo'HiooISHiSOOH‘zooo NEvent IILII 0 so 100 150 200 250 300 350 406 456 500 Er Figure 8.5: Jet50 Kinematic variables. Etotal: ET and ET before the selection cuts (dashed) and after (solid). 126 figure 8.5 continued. c) Jet50 Missing ET significance 6 T I I NEvent E 1 l 1 1 l 1 1 L 1 1 I 1 1 1 I :L 10 12 14 16 18 ~20 127 a) Jet70 Inclusive Jet PT 10’ 10° 105 10‘ 3 10 NJet N sod"eoo"" I IIIII I I IIIIIII‘ "I J 10 "Jet I I ililil lLllllll l l I I I I 10 -ILI I IIIIIIIIIIIII i r- P- i- 2 : m z 10 5 1o ; . s 55:3 0 500 1000 1500 2500 1‘3000 EMAGew b) Jet70 Mlsslng ET ‘rflrfinwfi'l'"'I'H'I'H'I'W'IH" E7 > m -. Z "‘3.” .....~u‘"'r._.s "F hi5" ..M. 7... . . n' I? La‘ru'yis'i‘J-r' . I ' iii")! IIIIIIIIIIII 0 50 100 150 200 250 300 350 400 450 500 ET Figure 8.7: Jet70 Kinematic variables. Etotal: ET and ET before the selection cuts (dashed) and after (solid). 130 figure 8.7 continued. 0) Jet70 Missing ET significance NEvsnt 131 a) Jet100 Inclusive Jet PT 10' NJet 1111l1111l1111 1111 111 1111 1 1° 0 100 200 300 400 500 600 700 PT(GeV) b) Jet100 Y (Rapidity) 105 _'_"'I""INT‘IHHI"W'TT'W'HI'H'I'H'I'fi' l. .‘-'._:J .:'--' .4 _ _I J {Mi-‘11. "--_ fl .. 5 :_:': 2% 1° 5‘ 5 : 7:. 4 : 5 1O _..11..111.11.1...11.. 1 l 1 [1.11111— -5 -4 -3 -2 -1 0 1 2 3 4 5 V Figure 8.8: Jet100 Kinematic variables: Inclusive PT , Y and ()5 before cuts (gushed) and after cuts on |Z| < 60cm, 0.1 < |Y| < 0.7, Etotal < 1960 GeV and £1 < 4 (solid). 132 figure 8.8 continued. c) Jet100 gin 20 I I I I I I I I I I I I r. r I r i—_ .. ,,,,, '-' - =:- - _: 'a ...--._.-; .5 so ”I” IllllllllllllllllflllllIlllll 16 14 12 10 8 NJet 6 4 2 o“? 133 a) Jst100 EnMl _L O b I lllllflri r NEvsnt —l O J lllllfll L 11111 b)Jst100 Missing ET IIITrTerIIIIIIIIIIIIIIIIITiIIIIII—IIIIIIIIIIIIII - I llllllll 1 1111111] Q“ ‘- ”Event 1 l llllllll 0‘4 0 ‘ ‘ (2" n ‘ ..(q:§"l.'v‘t 0‘ ' "5"." ll 111 1111 1111]111111111l1111l11111111111111l111 0 50 100 150 200 250 300 350 400 450 500 it Figure 8.9: Jet] 00 Kinematic variables. Etotal: ET and ET before the selection cuts (dashed) and after (solid). 134 figure 8.9 continued. c) JetiOO Missing ET significance ETTTIIIIIIIIIrTTTTITIIjIIjIITIIIIIIIIIII lllll I i [111"] l llllllll "Event 1. I lllllill l llllllll -.- -- I' |- .- 11111411111 LLIJIJIIILIIIIJIIII[PIIIJIII 2 4 6 8 10 12 14 16 18 20 ~ ET _L G [Hill O 135 8.5 Backgrounds Cosmic rays, accelerator loss backgrounds and detector noise were removed by cutting on the missing ET significance, ET =ET/\/EF_, where the sum is over all towers in the calorimeter. The following figures show scatter plots of ET versus lead jet ET and lead jet ET versus SET. The plots show the quantities before the event selection cuts, after the ET and after all of the event selection cuts. No cut has been made on the rapidity of the individual jets in the figures that follow. From the figures we see that the JethO sample is the most affected by the cosmic background. The event and jet properties of jets in the three highest PT cross section bins are scanned to make sure that we do not see cosmic events in the low statistics region of the cross section. 136 IIIIIrTIIIIIIIII—TIIII q —l d -i 93: (GeV) s llllllllllllllllllllllllllllIllllllllllllllllllr O '111'41111114 1111111111111111111111 100 200 300 400 500 600 700 800 900 1000 Lead Jet ET (GeV) Jet20 (b) — q q q q -—4 q l .. — d .1 — _- q u q q — .1 .4 c- 1 --1 . u: q q u—i a u q |- ET (GeV) alllllllllllllllllllllllllllllllllllllllllllllll IIIILLILJII ..i....1..1.|....|.1 00 100 200 300 400 500 600 700 800 900 1000 Lead Jet ET (GeV) Figure 8.10: Raw data distributions ET versus Leading jet for the jet20 trigger sample before selection cuts, after ET cut after all selection cuts. 137 figure 8.10 continued. Jet20 (c) 1 000 , er . . 900 800 700 600 500 400 300 200 1 00 00 100200 300400500 600 700 8009001000 Lead Jet ET (GeV) - .1 q q .1 .—i q d a: q — q - .4 .1 — 1 I '1 - .1 .- q u: —l q 1 .- q u .. q —-i :1 I q — n - d l- 55: (GeV) lllllllILlllllllllllllllllllllllllllllllllljill 138 Jet20 (a) 1000:....,....rm.1r...,....,.....]....,....1....1....__- 900%- ' - 8003— —f 9 7005— -: 01 600:— . a: 9 500;- " .g. — ul— 400: -2= 0 —f w E -, .4 : 300” --'/' -§ 200 1 100 —: 00 1 100”500LL500“bb0”600”000 70011050‘300'1000 Lead Jet ET (GeV) Jet20(b) 1000:'r'l"r'lT"'l""l'"'l""l""|""l"'rr' ': soc:— —; 800:- -Z 9 7°": ‘2 0 600; -'. ..- g 9 500E— _f if 400: .; N 300: a; 200 —: 100 :=' 00 ' 1003001500150 500 ébéuméuébb 900 1000 Lead Jet ET (GeV) Figure 8.11: Raw data distributions BET versus lead jet for the jet20 trigger sample before selection cuts, after ET cut after all selection cuts. 139 figure 8.11 continued. Jet20 (c) an. O .- q q . q —1 .. d .. .1 d .1 d .4 d _ q . .. d - .1 H . . ‘ .1 d .1 d _ d i I .. q - 1 l llll llllIllTilllllllllllilllll[IL 900 800 700 E 600 _ 9, 500 “a $33252”. J 400 '- N 300 200 100 00 ‘1‘66"é66“é66"i66"é66“é66”i66”666”666"1boo Lead Jet ET (GeV) llllllllllllllllllllllll]llllllllllllllllllld 140 Jet50 (a) 1000""1””1“"1"”1'"WU“IHWI'H'I'H'I'jg}: 900 ' —= 800 '~,}‘éu .. '5 700 . gut’fn- —; g 600 - . ,3” ' ' 5 g 500 l-'-;'.- "3 u: 400 . . -' ' 4: 300 .- ‘53. 200 ---.;.- -. _ —: 100 W‘.-;'.'.. —':‘ 00 100 200300 400 500 600 700 800 900 1000 Lead Jet ET (GeV) Jet50(b) 1000 l l INHIHHI'H'IHHI ': 900 -§ 800 -: 700 —§ § 600 —: g 500 —: u; 400 -= 300 —; 200 —: E 100 *V.‘ $4;- ' . {2’1'L111 . l 1111111 00 100 200 300 400 500 600 700 800 900 1000 Lead Jet ET (GeV) Figure 8.12: Raw data distributions ET versus Leading jet for the jet50 trigger sample before selection cuts, after ET cut after all selection cuts. 141 figure 8.12 continued. Jet50(c) 1000.1...,..1....,..1r.11....,...“”1111”,“: 900 .1 800 —§ 700 —: g 600 1 Q 500 —; at 400 1 300 1 200 1 10° . 1 00 100 200 sobfiibbus'oousbé '760‘1000'060'1000 Lead Jet ET (GeV) 142 )3 ET (GeV) 500 600 700 300 900 1000 Lead Jet ET (GeV) 1000 IIII IIIIIIIijIfiIIIIIIIIIIIIIIIIIIrTjIIIIIIlIII 2: ET (GeV) 0| 8 lllllllllLllLlJllllllllllJllllllllllllllllllllllq .1111111l....l....l..1.l....l....l.111l1...l1111 100 200 300 400 500 600 700 800 9001000 Lead Jet ET (GeV) Figure 8.13: Raw data distributions BET versus lead jet for the jet50 trigger sample before selection cuts, after ET cut after all selection cuts. 143 1mIIIIIIIIIITIITTIIIIITIrIIIII.IIIIIIIT.II'I_IIITIlII.:. o o a I .. 2: ET (GeV) 00 '160 260 360 400 560 660 760 8609601000 Lead Jet ET (GeV) i 11 — q —i d — _ _ 2: ET (GeV) 0| 8 lllllllllllllllllllllllIlllllllllllllllllllllll‘ 1 1111 1111 1111i.111111111111111111l1111l11 160 260 360 400 500 600 700 800 9001000 Lead Jet ET (GeV) Figure 8.13: Raw data distributions EET versus lead jet for the jet50 trigger sample before selection cuts, after ET cut after all selection cuts. 143 figure 8.13 continued. 2 E.r (GeV) § llllllllllllLlllllllllllIlllllllllllllllllllllll‘ IIIIII II IILIIII IIII 00" 100 '200 300 400 500' '600 700 300 '000 1000 Lead Jet E.r (GeV) 144 Jet70 (a) 551 (GeV) '_ 111;], 1111 1'1111'1 111111114111 00 100 200 300 400 500 600 700 600 900 1000 Lead Jet ET (GeV) cut I d _ q .1 q -1 cm -1 .1 d .1 _1 q .1 .1 q —4 .1 q q .4 —1 .1 1 u cl —1 -1 q j l :1 (GeV) § EllllIlllllllllllllllllllllllllllllIlllllllllllll 100 200 300 '400' '500” 600 700' '000 '900 1000 Lead Jet E.r (GeV) IPigure 8.14: Raw data distributions ET versus Leading jet for the jet70 trigger sample efore selection cuts, after ET cut after all selection cuts. 145 figure 8.14 continued. 51 (GeV) =1: - 1f'I’lv.1-1 11 l 1 1 1 00 100 200 300 400 500 146 llllllllllllllllllllllllllllllllllllllllllllllll‘ l I" l I I Ll LI I I I I I I I I I I I l 600 700 800 900 1000 Lead Jet ET (GeV) Jet70(a) 1000 ""I"fi‘l""l""I""‘*r"“1""I""'I'-'_:II:'I 900 ' '7’ 800 700 600 500 400 300 200 100 “o ”i66“é66“666”i66”566‘ié66"i66"666”é66”fooo Lead Jet ET (GeV) 2 Er (GeV) Illll11lllIlllllllll[llll'llllIlllllllll'lIll _III Jet70(b) 1000. .....,....,...11...,,...,...1......11..... 900 . 800 1., 7m .- .. 2 E1. (GeV) 01 8 alllIllllJlIlllllllllIllllllllllllllllllllllllll ILILIIAIIIIIILI II I IIIIIII 00"'100"200":i00' 400 500 600 700' 300 900 1000 Lead Jet ET (GeV) Fi b glue 8.15: Raw data distributions BET versus lead jet for the jet70 trigger sample eJ"C>re selection cuts, after ET cut after all selection cuts. 147 figure 8.15 continued. 2 E.r (GeV) 00' 100200 300400500 600 7008009001000 Lead Jet ET (GeV) 148 “""33'3x19‘. 9 a 9, at -. - ""1ug'tn‘;°ifl:n1u41'nnnn .: 1 00 10 20 304050 60 708090100 Lead Jet ET (GeV) Jet100(b) 9 E 0 : G -E v : "I “.2. . . .1 x101 100 Lead Jet ET (GeV) Figure 8.16: Raw data distributions 197 versus Leading jet for the jet100 trigger sample before selection cuts, after ET cut after all selection cuts. 149 figure 8.16 continued. Jet100 (c) :1 (GeV) 1 00 X1 01 90 80 70 60 50 40 30 20 1 0 10 20 150 llllllLllhllllllllllllllllllllllllllllllIlllllll ,_ 83 Q L 9.8 _F' ’58 tb 5.. X .5 °.. Jet100 (a) 1000* .. . . 1 . . . . 1 . . . .J...-, ... ...U , ...1 . I ...,...m , 1.14.... 900 ._ 1:...“- . _ .;. .12! r- f 800 700 600 500 2 51 (GeV) 300 200 1 00 I % IlIlIIIIlIIIIlIIIIIIIIIlIIIILIIIIIIIIIIIIIIIIIII 1002003004005006007008009001000 Lead Jet ET (GeV) Jet100(b) 1m HYU'IUIUV'UIIIrTIUI'IrJUZI".'TIII!II1IIIIIIIIIIIIIIII 900 1. - . .- 800 700 600 500 400 300 200 1 00 II %’lIIIIIIIILUIIIIIILIIIIIIIIIIIIIIIIIIll 100200300400500600 700 800'900'1'000 Lead Jet ET (GeV) ll 2 ET (GeV) Ellllllllllllljlllllllllllllllllllllllllllllll Fi b glare 8.17: Raw data distributions BET versus lead jet for the jet100 trigger sample efo‘re selection cuts, after ET cut after all selection cuts. 151 figure 8.17 continued. 2 ET (GeV) § 00 100200300400500600 152 llllllllllllllllllllllIlllllllllllllllllllllllllf 700 800 900 1000 Lead Jet ET (GeV) a) Jet1 00 :UT‘T‘YjYITTfIIII UIYIIIUTI‘ITYTITrFIrfir: — —1 -— —1 —1 fi — —1 1O — — u— — i- u—i b _- _ — P- —I a '- — 5 ._ _. > :‘-I —1 1 'I_ | 4 '-. -I- 10 _ , ._ __ : ,- - : '- 0' '3 —1 I— I "_ —1 I— I- ‘.' —l d C l- - -- Q '— -.- I--.- -I a 10 —- '- " :1 t: .- 1— -1 _IIIIIIILIII IlllIIIIIIIIIIIIIIIIIIIIIIId O 2 4 6 b) Jet70 10 12 14 16 18 N O 11! ..g rFTleIIIIIIIIIIYIIII'IIIIIIIIUIUUTIYI Fighte 8.18: ET before Samples. I l I I I l PII I I "10 12 14 153 I I l I I I l I I Ill!“ 1 llllflll 4 11111111 ”HI-L [1111111] 18 '20 ~ E1 16 and after the ET cut for the jet20, jet50, jet’70 and jet100 figure 8.18 continued. 1:) Jet50 _J 1 q .1 d J q -1 11mm] ilulml | NEvent d d d O O O to u oTI''l'ITllI—‘r'I'1'I1'I1Tl—'|"‘l'l'1'|1lq 1|"an [III p. I Z' . P I |- '- I _.. .' __ I I I l' 1' l h- '- I ,- ‘1 d) Jet20 MEvent d O 154 8.6 Raw Inclusive Jet Cross Section The raw inclusive jet cross section is constructed by joining the inclusive jet cross sections found for the four jet triggers together into a single continuous cross section. Previously the trigger efficiency and prescales were measured for the jet trigger sam- ples. The samples are combined by requiring that the trigger efficiency is greater that 995% and the prescales are used to scale the number of jets to that expected from the trigger path information. The inclusive cross section is defined as: AIY YdPTdY= AY det APT ' where N jet / e is the number of jets in the PT range APT corrected for trigger efficiency. f Ldt is the effective integrated luminosity including prescales and Z vertex correction. The rapidity range for the analysis AY is 1.2. The statistical uncertainty on the measured cross section is calculated as 6( .120 ) = 5(Njet/e) dPTdY detAPTAY dPTdY Where (5Njet/Njet) comes from counting uncertainty in a given bin, (66/6) comes from the trigger efficiency calculation and (6norm/norm) comes from the prescale Calculation. The uncertainty on the luminosity is common for all data points and will be treated with the systematic errors. 155 ‘1'” x 1/ (my-edits)? + (ac/e)2 + (Wm/norms, (82> PT bin N Jets Raw cross section (nb/GeV) Stat. Err. 61-67 12615 3.67348 0.00156761 67—74 8250 2.0592 0.00108662 74-81 110696 1.22883 0.000839406 81-89 73903 0.717842 0.00060013 89—97 42896 0.416662 0.000457217 97-106 65610 0.247133 0.000331986 106-115 38731 0.145888 0.000255073 115-125 25838 0.0875914 0.000187502 125-136 138485 0.0505314 0.000135788 136-158 132231 0.0241247 6.634316—05 158-184 55699 0.00859856 3.643366—05 184-212 21570 0.00309203 2.105328—05 212-244 8268 0.00103706 1.140526—05 244-280 3115 0.000347302 6.222696—06 280-318 1051 0.000111012 3.424286-06 318-360 322 3.077228-05 1.71487e-06 360-404 101 9.21341e-06 9.167689-07 404-464 38 2.542056-06 4.123756—07 464-530 2 1.216299-07 8.600489-08 530-620 1 4.459738-08 4.459736—08 Table 8.4: Raw inclusive jet cross section. Horizontal lines indicate trigger boundries. The Jet20 trigger is used in the PT range 50 — 74 Ge V, Jet50: 74 - 97 Ge V, Jet70: 97 - 125 GeV and Jet100 used for PT > 125 GeV 156 4 1ollllllllllllllllTIllllllllllll 3 10 .1 102 °. JIL=218|DH1 A 10 1.}. Ema 960 GeV > 3. MetSlg: sample dependent cut 0 1 ...,. .. Q 10'1 5°. |Z|<60 cm '9 '2 v . ' 0 1 c 10 . . ' '5 - Jet20 , E 10-0 0 Nb 10 . Jet50 Q * '5 -7 o Jet70 10 J 1 4, 10-0 . et100 10" Data uncorrected 10.10llllllllllllllllllllllllllJlll 0 1 00 200 300 400 500 600 P$°'(Gev) Figure 8.19: Uncorrected inclusive jet cross section as a function of PT, statistical errors only. 157 Run Event PT (GeV) Y EMF 143257 276943 401.184 0.398734 0.548158 145045 684636 412.519 -0.280769 0.12729 149387 4690452 437.213 -0.367935 0.354751 151555 1596656 411.366 0.100372 0.49213 152518 581053 457.819 0.116822 0.443789 153075 2273678 429.232 -0.65286 0.554674 153075 2273678 406.475 0.53179 0.5404 153374 1420764 400.446 0.260724 0.335107 153618 774625 455.659 -0.512773 0.966693 153447 4457374 410.003 -0.365903 0.467246 154208 885533 454.803 —0.407754 0.579744 154208 885533 415.526 0.237643 0.516058 155129 2465699 439.705 -0.282757 0.219059 155364 72998 549.186 -0.349703 0.826778 155997 1270890 403.824 0.233282 0.554097 156116 942464 416.51 -0.l25138 0.511041 161633 4356826 463.062 0.181779 0.477635 161714 580900 426.88 -0.65474 0.761816 161714 580900 401.172 -0.195772 0.623039 162423 7395708 413.256 -0.134176 0.553815 162423 406758 444.049 -0.273213 0.466517 162423 406758 403.282 0.277389 0.363556 162423 2703418 403.149 0.305488 0.539976 162396 1039796 461.466 0.499424 0.398409 162498 4411694 453.453 0.372285 0.468243 162631 890671 409.185 -0.642414 0.723598 162631 10531661 418.287 -0.469212 0.558444 163130 531768 442.902 -0.36703 0.348904 164989 2549548 443.148 0.32768 0.477917 165064 108597 423.084 0.40578 0.350614 166328 378124 439.903 -0.54438 0.117618 166662 5668262 401.974 -0.169606 0.331801 166662 5668262 401.986 0.560265 0.614567 167023 1813038 413.845 0.548813 0.418639 166927 8320832 449.278 0.225822 0.615626 166927 8320832 432.747 0.570658 0.593227 167297 1155843 497.889 -0.206923 0.361894 167954 2035694 433.498 0.166443 0.556618 Table 8.5: Jet properties for jets with PT above 400 Ge V. 158 Table 8.5 continued Run Event PT (GeV) Y EMF 168089 144763 415.191 -0.612085 0.870442 175066 151109 404.304 0.210371 0.471045 177316 2847553 413.749 -0.417775 0.200374 177316 2847553 415.868 -0.106148 0.388122 177339 1691542 442.926 -0.254262 0.694501 177339 1691542 432.401 0.298508 0.291503 177418 407378 475.227 0.284872 0.590199 177624 2485205 413.335 -0.223299 0.33967 178758 682071 449.035 0.206208 0.310963 178758 682071 436.161 -0.664422 0.579327 178921 82234 430.563 -0.604363 0.486315 166927 8320832 449.278 0.225822 0.615626 166927 8320832 432.747 0.570658 0.593227 167297 1155843 497.889 -0.206923 0.361894 167954 2035694 433.498 0.166443 0.556618 168089 144763 415.191 -0.612085 0.870442 175066 151109 404.304 0.210371 0.471045 177316 2847553 413.749 -0.417775 0.200374 177316 2847553 415.868 -0.106148 0.388122 177339 1691542 442.926 -0.254262 0.694501 177339 1691542 432.401 0.298508 0.291503 177418 407378 475.227 0.284872 0.590199 177624 2485205 413.335 -0.223299 0.33967 178758 682071 449.035 0.206208 0.310963 178758 682071 436.161 -0.664422 0.579327 17 8921 82234 430.563 -0.604363 0.486315 159 Run Event N Jets NQ12Zv Z (cm) ET (x/GeV) '1 ETotal (GeV) 143257 276943 3 1 -7.320 0.41908 1095 ._66 145045 684636 3 1 24.352 3.61982 1063.03 149387 4690452 8 1 5.662 3.82993 1082.47 151555 1596656 5 1 31.348 1.62779 1108.12 152518 581053 4 2 -7.958 0.371944 1 123.27 153075 2273678 5 3 2.595 0.384346 1355.99 15307 5 2273678 5 3 2.595 0.384346 1355.99 153374 1420764 3 1 42.643 0.307279 1 187.97 153618 774625 3 1 -10.838 4.54881 978.67 153447 4457374 4 2 13.980 0.303175 1 138.04 154208 885533 8 2 -7.697 1 .09069 1556.66 154208 885533 8 2 -7.697 1.09069 1556.66 155129 2465699 3 1 47.026 2.07085 1336.67 155364 72998 4 1 59.586 0.44107 1409.28 155997 1270890 4 1 -4.561 0.988793 1064.22 156116 942464 3 2 -46.612 1.71779 1142.24 161633 4356826 6 2 47.737 2.28901 1255.08 161714 580900 8 3 24.536 0.603169 1585.65 161714 580900 8 3 24.536 0.603169 1585.65 162423 7395708 3 1 4.765 1.43176 928.99 162423 406758 4 2 18.907 0.29795 1072.9 162423 406758 4 2 18.907 0.29795 1072.9 162423 2703418 2 1 31.102 0.394673 885.545 162396 1039796 6 2 8.199 0.283313 1 197.74 162498 441 1694 7 3 17.884 1.12226 1462.44 162631 890671 4 1 19.217 1.00378 1 124.75 162631 10531661 4 1 -41.588 0.14247 1351.46 163130 531768 5 1 ~16.563 1.41608 1160.1 164989 2549548 8 2 3.1 14 2.32176 1374.86 165064 108597 4 2 -50. 157 1.65477 1144.98 166328 37 8124 7 1 -2.406 1.4873 1182.97 166662 5668262 8 3 26.036 0.873594 1484.85 166662 5668262 8 3 26.036 0.873594 1484.85 167023 1813038 3 1 0.246 1.39682 1008.47 166927 8320832 4 1 27.501 0.912677 1299.43 166927 8320832 4 1 27.501 0.912677 1299.43 167 297 1 155843 2 3 13.239 0.843306 1246.78 167954 2035694 4 1 -18.591 4.01535 1010.95 Table 8.6: Event properties for jets with PT above 400 Ge V. 160 Table 8.6 continued Run Event NJets NQ12Zv 2 (cm) 13.1. (mew-1 Em“, (GeV) 168089 144763 6 1 37.156 2.76191 1362788 175066 151109 4 1 20.595 0.855962 1068.8 177316 2847553 2 1 6.539 0.308571 930.435 177316 2847553 2 1 6.539 0.308571 930.435 177339 1691542 10 2 11.709 0.522452 1544.84 177339 1691542 10 2 11.709 0.522452 1544.84 177418 407378 8 2 -25.408 0.650897 1538 177624 2485205 8 2 16.594 1.58629 1268.15 178758 682071 2 1 49.495 0.486888 1074.41 178758 682071 2 1 49.495 0.486888 1074.41 178921 82234 12 2 6.503 0.966076 1510 166927 8320832 4 1 27.501 0.912677 1299.43 166927 8320832 4 1 27.501 0.912677 1299.43 167297 1155843 2 3 13.239 0.843306 1246.78 167954 2035694 4 1 48.591 4.01535 1010.95 168089 144763 6 1 37.156 2.76191 1362.88 175066 151109 4 1 20.595 0.855962 1068.8 177316 2847553 2 1 6.539 0.308571 930.435 177316 2847553 2 1 6.539 0.308571 930.435 177339 1691542 10 2 11.709 0.522452 1544.84 177339 1691542 10 2 11.709 0.522452 1544.84 177418 407378 8 2 -25.408 0.650897 1538 177624 2485205 8 2 16.594 1.58629 1268.15 178758 682071 2 1 49.495 0.486888 1074.41 178758 682071 2 1 49.495 0.486888 1074.41 178921 82234 12 2 6.503 0.966076 1510 161 Chapter 9 Jet Corrections 9.1 Introduction We wish to make a measurement of the hadron / parton level inclusive jet cross section as a funtion of jet PT. In an ideal detector we could measure the jet PT’s in every event and the cross section would be given simply by normalising the resulting histogram for acceptance and luminosity. However, we do not have an ideal detector and the determination of the inclusive cross section is complicated by several efl'ects: e Background events from cosmics, detector noise and beam halo are present in the raw data. e Not all jets are observed in the detector due to finite efficiency of the detector for hadron level jets at the calorimeter level. 0 The measured transverse momentum Pig“! is smeared due to finite resolution of the detector. This smearing will cause migrations of jets in PT. e The measured PT of a jet underestimates the original hadron level PT by up to ~ 20%. Like the previous point, this effect also leads to PT smearing of the true hadron level cross section. 162 The background events are removed by the selection cuts applied when constructing the raw cross section (see section 8.5). No further correction needs to be made for background after that. The efficiency of the calorimeter for detecting jets was studied in Pythia. It was found that for hadron level jets with PT above 25 GeV the detector is 100% efficient, i.e. there will always be a matching calorimeter level jet. To facilitate comparison of the results to theory or other experiments we need to remove the detector effects. After the detector effects are removed we have a measure of the hadron level inclusive jet cross section. For comparison to NLO pQCD an additional correction to remove the underlying event contribution to the energy in the jet cone and the fragmentation losses from the jet cone will also be needed. The jet corrections employed to remove the detector effects are based on Pythia events that have passed through a detector simulation. The calorimeter to hadron level jet corrections were made in two steps: e Average P14“ correction to remove the energy losses of hadrons going through the calorimeter; this is applied on a jet by jet basis. 0 Bin by Bin unfolding to remove the effect of PT smearing. In the following chapter we outline the Monte Carlo generation, weighting and the jet correction procedure. 9.2 Monte Carlo Simulation Jets are generated using Pythia 6.203 and Herwig 6.4. QCD (2 —+ 2 processes) includ- ing initial and final state gluon radiation as well as secondary interactions between beam remnants. The jets are then passed through a detailed detector simulation (CDF SIM). The parton interactions are generated using LO QCD matrix elements. 163 The PDF set used to describe the proton and antiproton is CTEQ51. This PDF set contains the CDF and D0 jet data from Run 1. The Pythia samples are generated using a tuned set of parameters that control the soft gluon emmision. The parameters were tuned to make Pythia reproduce the Run 1 underlying event properties. 9.2.1 Weighting the Monte Carlo The Monte Carlo is generated using a number of [ST thresholds, where 151‘ is the tree level hard scattering momentum. The events are generated with an event weight and a fixed tree level cross section for the sample (0,3744... 2 0(I3T 2 1314'! M)). The samples can be joined into a continuous spectrum by applying to each event/ jet the associated event weight. Figure 9.1 shows the number of events in the sample as a function of the tree level momentum PT of the samples used in the following work after weighting. For both Pythia and Herwig we use 500K events from each of the PT samples; PT =18,40,60,90,120,200,300 and 400 GeV. Figure 9.2 show the inclusive jet PT distribution after the samples have been combined by weighting. The lowest 1571 threshold used in this study is 18 GeV, therefore the lowest 131an that can be used for this sample while still avoiding the generator level threshold bias is 35 GeV (figure 9.5). Due to trigger efficiencies in the data, the inclusive cross section measurement begins at an uncorrected PT of 50 GeV . This data threshold has a corresponding threshold in corrected PT of 58 GeV. The corrected cross section will be presented with a lower corrected PT bin edge of 60 GeV. The calorimeter level inclusive jet distribution in figure 9.2 has two spikes/ bumps around 320 —— 350 GeV. These features come from jets that were generated from a 164 sample with a low 15744 in (high weight) that fluctuate to a high PT after passing through CDFSIM. These spikes/bumps do not have any significant effect on the corrections. 165 d a ITIIIIITFTIITTITITIIIIIIrYIIIIIIIIIrj did ad" d—L—L—L OOOO d—l COO O Mama/(2 GeV) M “abdusa-‘fl O O d-h—L dOOO LIIIIIILJIIIIJIIlLlIIIIIllIIlIIlI 0 100 200 300 400 500 600 700 800 P?“ (GeV) Figure 9.1: Number of events as a function of ET for Pythia after weighting. The smoothness of the plot indicates the weights have been found and applied correctly giving one continuous sample with a single threshold of PT = 18 GeV coming from the lowest of the samples. 166 [III—[UITIIUIITI'IIlTIjrfIlIIIIIIIIT '- I a _ I ‘ dd N“ e I e d d Calorimeter Jets d c d-I-l-L OOOO ----- Hadron Jets Q 6 Q Q ‘ 0 .Q 9 Q N Jets[(2 GeV) . ’5 ‘0 § 9. .. Q ‘- N a b "I O N O 0 I , I d-L—L-l-L-L-l—L—L dOOOOOOOOO llllILLlIlllllllIIJIIlllllIIIII 0 100 200 300 400 500 600 700 PT (GeV) Figure 9.2: Inclusive Pythia PT distribution for hadron level and calorimeter level jets after weighting the events. The spikes/bumps around 300 GeV arise from jets from a low 1311 (larger weight ) sample fluctuating into a high PT bin. They are associated with CDFSIM and therefore are not present in the hadron level distribution. 167 9.3 Average P74“ Correction The average leet correction is applied to the transverse momentum of each jet in order to compensate for the energy lost in the detector on average. The correction is derived from the Monte Carlo in the following way: e The MidPoint jet algorithm is used to reconstruct jets at both the hadron and calorimeter level in the Monte Carlo. At the hadron level, 1 GeV seeds are used and no other requirements are made on the particles before clustering. At the calorimeter level, the standard clustering cuts / requirements were used. e Hadron and calorimeter jets are matched into pairs. Jets are considered matched in (Y, ()5) space if their separation AR = JCSYV + (A¢)2 is less than AR = 0.7. Figure 9.4 shows AR, A65, AY and P1151 “d/ PIC-7"“ for the matched jet pairs for Pythia. The resolution of Y and (b are very good and no correction is required for these quantities. The PT of the jet is underestimated by ~ 20% at low PT improving to ~ 10% at high PT. It is this scale that we wish to correct for with the average PT,“ correction before unfolding. 0 From the Pythia jet pairs, the calorimeter PT is fixed in 5 GeV bins; from the corresponding hadron distributions we find (PTIet(Had)). This correlation is fit to a polynomial : (PIHad)=A+BxPT+CxP%+DxP_%+ExPg. (9.1) e The fit range is determined by the thresholds in the Monte Carlo sample, for the weighted sample this is the threshold associated with the lowest RTM'n sample used. 168 9.3.1 Calorimeter Level Thresholds at Hadron Level In the data there is a threshold (lower bin edge) from which all data below is disre- guarded. The location of this threshold is set by the trigger efficiency of the Jet20 sample. When a jet by jet correction is made the lower edge of the bin in uncorrected PT is moved to a higher value in corrected PT. When considering the inclusive jet distribution that has the Average leet correction applied we must ensure that our first bin is away from the corrected PT threshold. This prevents us having any bins that are underpopulated due to the calorimeter level trigger efficiency cut. 5111(1) bln(l+1) bln(l+2) bln(l+3) 0 Raw jet . Jet Corrected for P?" Cut P,>naw(P;"")—> bln(l-1-2) has a Corr(P¥'") threshold. Only conslder corrected 0 ..... - ............. - 7; t. i " ' dlstrlbutlon for Corr(P,)>bln(I+3) 0 >9 Raw(P¥_'") 'Corr(P¥_'") Figure 9.3: Avoiding seeing the trigger efficiency cut in the P141"3 inclusive jet cross section. 169 UIWTTIrIIlIIYlTYr'TrY—IIIIIFIUIYIrrfiT NEvents § ngIlLlllllllllllllLllLlllllllllllllllJJllli OIILIIIILJIL 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 I}?! Ca? I P, [PT 7 11 II'YITIITIIIITfIIIIIlIllIIIIIIUI'IIIIIIIIIITWTI 3000 I l 2500 2000 1500 "mm. 1000 500 llJllllllJllIlLLJlllllllllllll Sl1.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 YHad_YCal Figure 9.4: Pythia P1}? ad/ P]? “I, YHad — Y0“, and (13H “d — 450‘“. From the plots it is clear the Y and 45 resolutions are good, however the jet PT is underestimated by ~ 20% with respect to the hadron level. 170 figure 9.4 continued. 7 10 35m IUUIIIU'TV—TTIIIII[ITIT'ITIIIITIITIIIII'Y'IIIUI' 3000 2500 2000 "mm 1500 1000 500 lllLlllllllllllllllllllllllllllllll. 31.5 -0.4 -0.3 _ -0.2 -0.1 0 0.1 0.2 0.3 O b 2:: ¢Hed_¢ . q _1 .7 .. .. fl .1 1 .. _. .1 .1 .. _. q u .. 3000 2500 2000 1500 1000 "sum o l l 1. D 3 ‘ITIlllJlllllllIllllllllllllllllllllllJJll-I' 171 UIFITIII[ITIIFIYIIIIIITIIFIITIIIIIITIUIIIrrT—[Illl A 01 Ill 1111 l I I llllllllllll lllllll IP?"> llllllllllll?lll+ 0.8 o Pythia P}>18Gev weighted— 1 . PythiaP‘,=10 I 0.6 _ r1111[lllLllIljlllLLllllllllllilllllllIJIIIIIIIILI- 10 15 20 25 30 35 40 45 50 55 60 Calor PT (GeV) Figure 9.5: It is clear there is a bias in the combined Pythia sample below 30 Ge V. The bias is seen as a difl'erence between the (Pygad/qual) for the PT 2 10 and all other samples with RT _>_ 18. This bias is due to the generator level cut in the RT 2 18 sample which is the only threshold present in the combined sample. 172 P-lrtadron (G ev) 100 80 60 . = f 1": 40 - _ -_ 20 _ - — o ' ' I; " '1" F 0 20 4O 60 80 100 P?” (GeV) Figure 9.6: Using the previous plot to set the lower limit of calorimeter PT, we see that we do not truncate the tails of the hadron PT distribution in the following work. 173 700 .- I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I W— - p0 = 1.5971 10.57 - _'_ p1 = 1.1741 0.254 j 500 _ p2 = 000041141 0.001762 __ _ p3 = 46886-071 4.5326-06 - : p4 = 4.0736101: 3.849-09 : 500'— l A - _ > - _ d) : 2 S2400: : 5 ; ; 200:— -{ I o Wad-P?" correlation (Jets matched in (Yip) ) Z 100 _-_- —_ : —Fit: p0+p1(P?°‘>+p2 35 GeV which is set by being in a region where Pythia 18 sample is unbiased. 174 9.4 Smearing Correction 9.4.1 Pf)“ (Corr) Resolution and Bin Size The smearing correction is applied to the average P71“ corrected cross section on a bin by bin basis. We require the cross section bin sizes to be greater than or equal to the PT resolution of the jets being measured / corrected. We measure from Pythia the PT resolution of the calorimeter to the corrected jets. The resolution (0(fb) ) is found as a function of qual (Corr) by fitting gaussians to the fb distributions associated with fixed intervals of P5? “1(Corr). The quantity fb is defined as: _ Pf!“ (Corr) — P7206! - 13th T fl) 1 (92) where Pfaal(Corr) is the PT of a jet that has had the average PT,“ correction applied to compensate for energy lost in the detector. The mean of the fit (figure 9.9) gives a consistency check on the hadron scale correction. It should be close to zero: it is expected to differ from being exactly zero since the correction is found using the full distribution where as the gaussian used in the resolution study truncates the tails of the distribution. The standard deviation 0(fb) is the resolution of the calorimeter to corrected jet PT (figure 9.10). The resolution multipled by the PJQGI(Corr.) gives the la bin width in corrected jet PT. 175 53>> I « IIIIIIIIPIIJIIIIIIIIIIIIIIIIII 100 200 300 400 500 600 P$‘“(Corr) (Gev) .5 d O1H Figure 9.9: mean(fb) ’s from gaussian fits to fb distributions (e.g. 9.8 ). This value acts as a consistancy check for the Pfh’e correction. We see the mean values are al- most 0.0 which is what we expect if the correction is reasonable. The actual correction is derived from the statistical mean which includes the asymetric tails of the hadron distribution. The first unbiased PYQ‘WC'or) bin in the Monte Carlo sample is 45.0 Ge V. Below this value we see the effect of the bias coming from qual. 177 002 j I I I I I I I I I I I I I I I I I I I I I I I I I I I I I -1 0.18 0.1 6 0.1 4 0.1 2 0.1 0.08 0.06 0.04 0.02 1 — 0(fb) p _ r -..I-III" IIIIIIIIIIIIIIIIIIWIIIIIIIIIIIIIITI III 0 IllllllllllJJlllllllllPlLlllll 1 00 200 300 400 500 600 P$“(Corr) (Gav) 0 Figure 9.10: 0(fb) from gaussian fit. This is PT(Corr) resolution which is used to set the bins of the cross section. 178 I 50 I I I I I I I I I I I I I I I I I TI I I I I III I I I llllll (1(fo P$“(Corr) no a: [WIIIIIIIIIIIIIHIIIIIIIIIIWITIIIIIIIIIITI llllllllllll ml .1 o I I I I I I I I I I I I I I I I l I I l I I I I I I I I I I A 1 00 200 300 400 500 600 P$“(Corr) (Gav) 0 Figure 9.11: o x qual(Corr) gives the bin widths in GeV. The bins are chosen to be 10 for Pg‘dw'orr) < 100 GeV, 20 for PIQ‘“(Corr) < 200 GeV and 2 30 for PTQ‘WCorr) > 200 GeV depending on statistics. 179 9.4.2 Bin by Bin Smearing Correction The average P11“ corrected jet cross section must still be corrected for the smearing effects of the calorimeter to give a hadron level inclusive jet cross section free of detector effects. The smearing correction is done with the following prescription: 0 Jets are reconstructed at the hadron level using the HEPG final state particles. The ratio of the true hadron level cross section and the average P11” corrected cross section is taken. No matching requirement is made on either of these cross sections. The ratio of these cross sections gives the bin by bin correction factors: [Iadron ._JEL . . Ci“ NCO“)? Ibm(z)a(9-3) Jets where N {ggdrm is the number of jets in a bin of the hadron level distribution and .Iet N Calm is the number of jets in a bin of the average PT corrected distribution. Jets All selection cuts are applied to the measured distribution; however only the rapidity cut is applied to the hadron level jets. 0 Figure 9.12 shows the bin correction factors for Pythia (” smoothed” and raw) that are applied to the data. The bin by bin corrections derived from Pythia are fitted to a smooth curve. The curve is then integrated over the bin and divided by the bin width to give a smoothed bin by bin correction. All figures where bin correction factors are shown include the original Pythia points with statistical errors and the smoothed correction. The statistical errors from Pythia were propagated into the cross section statistical error. 180 I I I I I I I I I I I I I I I I I I III IfirT Ifi fiI I I I I r1 1 2.4 -- —- : o Pythia Smearing Corr. : l— a g 2'2 L— — Smoothed Correction #2 —_‘ z: _ 4 0 2 —- — 1‘4 l3 : 0 1.8 F— + —‘ O ._ _ m _ J ,5 1.6 :- -"— L h - . —1 s : = E 1 .4 —- _ a) C I 1.2 :— —‘ 1 _ I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I q 0 1 00 200 300 400 500 600 700 PT (GeV) Figure 9.12: Bin correction factors for Pythia. These correct the average PT,“ cor- rected calorimeter distribution for the smearing associated with detector resolution. The histogram is the smoothed correction that is applied in the data. The statistical errors from Pythia were propagated into the final cross section comparisons. 181 9.4.3 Parton to Hadron Corrections The corrections thus far describe how to correct the inclusive jet cross section to the hadron level. We must correct the cross section for hadronisation effects (frag- mentation and underlying event) in order to make a fair comparison to N LO pQCD. These two contributions act in a competing manner. The hadronisation process acts to reduce the amount of energy clustered in a hadron level jet with respect to the original parton level jet. This effect is typically small in the PT range for which we are currently measuring the inclusive jet cross section. The underlying event acts to add energy into the jet cone at hadron level. This energy is coming from beam-beam remnant and soft spectator interactions. As we go to lower PT this effect becomes much larger that the hadronisation effect. The hadron level to parton level correction is derived from the Monte Carlo. As with the calorimeter to hadron level corrections we derive the corrections using both Pythia and Herwig. However, we correct the data using the correction derived from Pythia as this provides a consistant correction scheme. The full parton to hadron level correction is given by hadron(UE) 0' 05"” = 1 (9.4) rton no-UE) ’ or ‘ hadron.(U E) where 0i is the hadron level inclusive jet cross section in bin i with un- ofartm(no_UE) is the parton level inclusive jet cross section, without derlying event, underlying event. The fragmentation and underlying event corrections were also stud- ied separately using: 182 ahadron(U E) f Uhadron(no—U E ) UE _ i _rag _ i C” _ hadron(no—U E) and Cl _ aparton(no—U E) ’ (9'5) ”1‘ 2' hadron(no— U E) hadron(U E) and Uparton(no—U E) . i where o are defined as above, a is the hadron level cross section without underlying event. Using these definitions we see that Off-m = CiUECifmg [36]. We should note here that Pythia and Herwig imple- ment these effects is different ways. Pythia uses a multiple parton interaction model which contributes to the underlying event of jets at parton and hadron level. Her- wig uses a beam-beam remnant model that only contributes to jets at the hadron level. When deriving these corrections with Herwig, the underlying event correction is underestimated and therefore so too is the full parton to hadron correction. For Pythia we see that for PT > 60 GeV the underlying event correction dominates the full hadron to parton correction. We present comparisons between NLO pQCD and both the inclusive jet cross section corrected to parton level and also corrected to hadron level with the underlying event correction. We take as a systematic the difference between the Pythia and Herwig parton to hadron level corrections. This is a conservative estimate given that pythia is tuned to reproduce the CDF run 1 underlying event properties. 183 Pythia Paxton-Hadron Corrections 1.5_I I I I I I I I I I I I I I l I I I I I I I I I I I I r1 I I I 1.4; L c e _ o _ _ :3 : I g 1.3— — c'» 1‘2”: l+ 1 E : *l : ll; _ - D a ++H++ _ 1'- +—I——l— 1 l i l + d ' ‘ ‘ ‘T‘W— ' l ' . LI I I I I I I I I I I I I I I l I I I I I I I I I I I I I I I 0 100 200 300 400 500 600 PT(Gov) Figure 9.13: Fragmentation + underlying event correction for Pythia. 15,1'”1""!""I""l""l""l".. ...a .5 51.4%} —; 31.3:— —: '5'”? in ‘1 ...L + _= i +++++ +_+_ —+—3 1: +.—+— r— - ....1.LLLJ....1....J....1....1.; o 100 200 300 400 soo soo P,(Gov) Figure 9.14: Pythia underlying event correction. 184 Frag. Correctlon 1.“ I I I I .. l l I I I I F '_ 1 - 1- E + +++_+_—tl——+—- 3‘ 0.95 — HHH> _ 0.9 '— .2 035 _L _: 0.8 l- 1 L L I I I 1 I 1 I 1 I I 1 I n 1 I 1 I 1 . L 1 l . 1 . l l 1 L 0 100 200 300 400 500 000 PT (Gov) Figure 9.15: Pythia fragmentation correction. 185 Herwig Parton-Hadron Corrections 11_....I....,....l....,....,........._ 5 1.08; - Herwig Fragmentation+ Underlying Event Corr; *0 i— .4 o ; Smoothed Correction : g 1.06— — 8 : I m ”’4.— t a - _ c 1.02— — ° — — '5 I ’— I 1 g '_ 1 l 1 ' l l 7 _ ‘1’ I I E 0.98— — m _ _ E : : II- 0.96: _- 0 100 200 300 400 500 600 700 PT(Gev) Figure 9.16: Fragmentation + underlying event correction for Herwig. 1.2_....T....I.r..l...rl....l....l.... : - Herwig Underlying Event Corr. —¢ 1-15 :' Smoothed Correction j 1: : : o _ —4 E 1.1— _. o I I O 1.05— — Iu - . 3 I 2 i— — 1 : I i. _ 0.95 (i 100 200 300 400 500 600 700 PT(Gev) Figure 9.17: Herwig underlying event correction. 186 PT (GeV) Had Cor Stat Err UE Cor Stat Err Part Cor Stat Err 61-67 1.347 0.036 1.372 0.049 1.222 0.033 67-74 1.321 0.035 1.284 0.048 1.182 0.035 74-81 1.298 0.038 1.243 0.050 1.130 0.038 81-89 1.280 0.041 1.248 0.050 1.157 0.031 89—97 1.265 0.025 1.165 0.036 1.085 0.021 97—106 1.253 0.016 1.153 0.028 1.070 0.022 106-115 1.244 0.018 1.149 0.032 1.070 0.026 115-125 1.238 0.017 1.141 0.032 1.067 0.026 125-136 1.233 0.019 1.123 0.033 1.058 0.026 136-158 1.231 0.013 1.114 0.025 1.052 0.020 158-184 1.234 0.013 » 1.100 0.022 1.046 0.018 184-212 1.245 0.014 1.065 0.024 1.016 0.023 212-244 1.264 0.014 1.046 0.019 1.007 0.018 244-280 1.293 0.014 1.054 0.022 1.007 0.021 280-318 1.336 0.017 1.023 0.021 0.992 0.019 318-360 1.397 0.022 1.030 0.019 0.993 0.020 360-404 1.482 0.023 1.041 0.023 1.007 0.021 404-464 1.618 0.019 1.036 0.024 1.002 0.018 464-530 1.840 0.031 1.035 0.030 1.007 0.025 530-620 2.225 0.047 1.046 0.020 1.014 0.018 Table 9.1: Summary of Pythia bin correction factors: Had. Corr are the calorimeter level to hadron level correction factors, UE. Corr are the underlying event corrections and Part. Corr are the full parton level to hadron level correction factors. 187 PT (GeV) 00350") (nb/GeV) 500919”) o(had) (nb/GeV) 60(had) 51.57 3.54393 0.00239073 11.7442 0.304023 57.74 4.77715 0.00155505 5.23333 0.159222 74.31 2.71935 0.00124332 3.55377 0.102557 3139 1.55533 0.00033331 1.92301 0.0533199 3597 0.394051 0.000559753 1.10337 0.0219453 97.105 0.525322 0.000434255 0.552332 0.0034304 105-115 0.311059 0000372457 0.339707 0.0055032 115125 0.134451 0.0002721 0.225542 000324352 125135 0.109075 0.0001995 10135035 000204021 135153 0.0514123 9.53501505 0.0540773 0000579477 153.134 0.0134153 533203505 0.0227453 0000242952 134.212 000542575 3.035505 000301434 9.75035505 212-244 000225774 153232505 0.0027944 370324505 244.230 0000723145 3.9792505 0000951751 1.57152505 230313 0000242093 505579505 0000321771 733957505 315350 7.253505 2.53457505 0000101243 3.99794505 350404 1.90554505 1.31373505 2.73233505 1.97373505 404-454 4.54393505 5.51539507 7.40197505 901973507 454—530 103335505 2.50745507 1.9109505 4.54533507 530520 4.45973503 4.45973503 9.79972503 9.30193503 Table 9.2: 0(P7QO") is the average Pilaf corrected inclusive jet cross section. o(had) is the hadron level inclusive jet cross section (before UE or fragmentation corrections). 188 Chapter 10 Comparison of the Data to Pythia+CDFSIM 10.1 Introduction The raw data is corrected for hadron scale (average P%et correction) and smearing using Pythia Tune A plus the detector simulation (CDFSIM). While the ultimate com- parisons are to NLO pQCD predictions, which contain at most three partons in the final state, here we make some comparisons of the data to Pythia Tune A+(CDFSIM) using CTEQ51 PDF’s. The fragmentation / hadronization of partons is well modelled for L0 QCD predictions. Pythia uses LO matrix elements, plus a leading log approx- imation for the parton shower, then applies a string fragmentation model to convert partons into particles. The resulting particles are passed through the detector sim- ulation. We compare Pythia to the four trigger samples in a region where both the data and Pythia are away from trigger and generator level thresholds. Pythia 18, 40 and 60 samples are used in the comparison to the Jet20 data sample. Only events in which the lead jet PT > 50.0 GeV are used in the comparison. Pythia 40, 60 and 90 samples are used in the comparison to the Jet50 data sample. Only events in which the lead jet PT > 75.0 GeV are used. Pythia 60, 90, 120 and 200 samples are used in the comparison to the Jet70 data sample. Only events in which the lead 189 jet PT > 100.0 GeV are used in the comparison. Pythia 90, 120, 200, 300 and 400 samples are used in the comparison to the Jet100 data sample. Only events in which the lead jet PT > 150.0 GeV are used in the comparison. 10.2 Quantities of interest Figures 10.2 and 10.3 show the ET and E} comparisons for the jet trigger samples and Pythia. These quantities are sensitive to the simulation of both hard and spectator interactions. The inclusive jet analysis uses a cut on E} to reject background events. The MC distributions imply that ~ 1% of the good events are rejected. Figure 10.4 shows the transverse momentum difference in the two lead jets in the event (APT). This difference can result from: energy resolution of the detector and additional jets produced from the hard scattering. The agreement in this plot suggests the resolution and jet multiplicity are well modelled. Figure 10.5 shows the difference in azimuthal angle between the events two leading jets (A03). Like APT this quantity also depends on the number of jets in the event and resolution non-uniformities in (f). Again good agreement is observed. Both APT and A43 are sensitive to detector resolution and additional radiation but these quantities do not distinguish between the two effects. The effect of additional jets (QCD radiation) can be minimized by measuring the energy or momentum mismatch parallel to the axis defined by the leading two jets (KTH) (this work is included here for completness only, a more detailed discussion can be found in chapter 7). The direction of the parallel axis 11 is defined as the perpindicular bisector, t, of the two jets: fil + 112 t: —— 10.1 |fi1+fi2| ( ) 190 where {11,2 are the unit vectors along the two leading jets in the (a: — y) plane. From this KT” is defined as [3.1.1.11 — 13T2.n. Figure 10.6 shows the KT“ comparisons for the jet triggers normalised to the average jet PT. The good agreement indicates the jet resolution is well modelled by the simulation. The momentum imbalance along the t direction, Figure 10.7 shows KT; which is sensitive to both energy resolution (non-uniformities in (b) and to additional jet production. The CDF calorimeter measures the energy in two depth segments. The first (closest to the beam line) is the electromagnetic compartment. The second is the hadronic compartment. The electromagnetic calorimeter measures the electromag- netic particles (mainly iro’s) in the jets, along with some energy from hadronic par- ticles. Figure 10.8 shows the fraction of the energy deposited in the electromagnetic compartment for jets. Figures 10.11 and 10.12 show the comparison of the inclusive jet cross section between data and Pythia. In the plots the data has been corrected for multiple interactions, hadron scale and smearing. Both the data and Pythia have underlying event still present. On these plots the 5% systematic uncertainty on the energy scale (see discussion in chapter 12) is indicated by a shaded band. The Monte Carlo has been weighted according to the luminosity (as described earlier) to include Pythia samples with PT 13, 40, 50, 90, 120, 200, 300 and 400 GeV. Table 10.1 gives the cross section values and the bin correction factors for each bin. In figures 10.1- 10.10 both data and Pythia have been normalised to a unit area. 191 10.3 Results NEvents lllllllIlllllllllllllllllllllllllllllllllllllllll 350 -10 ' -50 0 50 00 1 0 Z(cm) Jet50 0.05 I I I I I I I I l I I l l I I l l l l l I l l T#: 0045 . -§ 0.04 1 ‘ . . —§ 0.035 1 " —§ g 0.03 . —; 2 0.025 . _: III : Z 0.02 - —; 0.015 ' —§ 0.01 —: 0.005 —: I 4 . 1 . . . . 1 . . . 7- 350 -100' -50 0 50 100 150 Z(cm) Figure 10.1: Z vertex distributions for data (histogram) and Pythia (points). 192 figure 10.1 continued. NEvents o o N as lllllllllllllllllllllllllllllIlllllllllllllllllll NEvents 193 I I I I I I I I I I I I 10.1 I I I I I r I I I I I I I I 10'2 ' - ‘- 10" “'~ E I “a _ zIIII 10 Q i 10'5 - r l -0 ’f 10 II '7 I I L 1 I I I l I I I 1 I l 1° 0 20 40 60 t100 I, M50 10-1 ' ' ' I ' ' I I ' ' ' l I ' ' T I ' 10'2 ' "-~ _ £104 " ' _ I 4 O I m 10 I z .5 ”I? i L It 10 I W 7 10‘ l '7 I I I 1 I I I 1 I I I 1 I I I i I 1° 0 20 40 60 so 100 I, Figure 10.2: E distributions for data (histogram) and Pythia (points). 194 figure 10.2 continued. JeflO 10 - III .5 10.7 7 144 l 1 111 l 1 L l l l L l 1 1 l l 0 20 40 60 80 100 120 140 Jet1m 1 195 .1020 "Events 4 l l 1 l? 6 a 10 ET Jet50 _fi f I fi‘ I I j I I: r i -1 10 E __ .2 - d g 10 g _3 _ 10 4 7 L l_ L l 1 P1 1 l 4 17 1° 0 2 4 6 a 10 51' Figure 10.3: ET distributions for data (histogram) and Pythia (points). 196 figure 10.3 continued. Jet70 I I III“ 3 -2 S 10 2— Ii 5 z - 10.3 :— 1 l lllllll l llllllll Jet100 HI— 0 1 llllllll I llIlll g 10.2 5— fi 5 z : 10.3 E— T‘ l llllllll lllllllll l llllllll —L O C N 197 10.1 TIIIIII'ITTrITU'IIU'III'UIIIIITTTIITII ‘° ‘1 10.7 IIIlIII1III1III1III OOHHN ‘1 I11III1III o 20 40 120 100 200 APT 1o ITTIIIrIIIlIIITjIIIIIIlIII[IITITIIIFI 10 1111111111 11 11 111 lljllllLJll 0 20 40 60 80 100 120 140 160 180 200 APT Figure 10.4: APT distributions for data (histogram) and Pythia (points). This quan- tity is sensitive to radiation and non-uniformities in detector ET resolution. 198 figure 10.4 continued. JOUO M100 "Emu 'T‘ I I l l' l l l ‘1 I." 0.9? 911 v t 'x 0 I. lllllllllllllllllllllll 20 40 60 80 100 120140160 180 200 API. IIII'I II'II I'fIIlIIIITIIIIIIIIIIIIII “I [IrI it, "‘ I: t + II? J l l I l l l l l I l I l l l l l I l l l I l l l l l l l l I l I I 1": III 20 40 60 80 100 120 140 160 180 200 A PI. 199 1 VIIIIIITf'IIII'IrIIllTIIIIIIIIij'lIII q — _L O 61 0 L F A¢ Jet50 10'1 10 £1043 10 10"5 3 ' T 10 I I ,III1IIII1IIIL1IIII1I IIIIIII1f 0 0.5 1 1.5 2 2.5 3 3.5 4 A4) Figure 10.5: A05 distributions for data (histogram) and Pythia (points). This quantity is sensitive to radiation and non-uniformities in detector ()3 resolution. 200 figure 10.5 continued. Jet70 111111 7 I l l l l I l l l I l l l l I l I l l l l l 7 1° 0 0.5 1 1.5 2 2.5 3 3.5 4 A ‘1) IIIIIIIIIIIIIITIIIIIIITITTTII‘IT a". ' 0.5 p ”b h 1 '1.5””2'”'2.5'” A¢ 201 Jet20 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 X ..L ‘°. NEvents lllllllllllllllllllllllllllllllllll *lllllllllllllllllllllllllllllllllllllf .(3 vw—wv I l 1 1 -l llll 3- O 0 0| .5 KT Parallel u‘ "‘3. IIIlllllllllllllllllIllllllllllllllllllllllllll IK ‘1" .5 KT Parallel Figure 10.6: KT“ distributions for data (histogram) and Pythia (points). This quan- tity is related to the detector ET resolution. 202 figure 10.6 continued. Jet70 x10'1 0.5 ' * P h llllllllllllllllflfllllll -l1111l1111l1111lllllllllll- KT Parallel —L ‘°. l 1! ‘Llllllllllllllllllllllllll O N dIIIIIIIIIIIIIIIIIIIIIIIIII l 1 KT Parallel 203 Jot20 0.25 I I I I I f I I I I r T I I I I I I r I '_I t : 0.2 _ :- -— 4 -— q - 1 g 0.15 + 1 z... I I 0.1: —_ _ -1 0.05 _— j I— —I F- _ i— — cP KT Perp I I I I I I I I I l I I I :_ _n" 00 0.1 0.2 0.3 0.4 0.5 KT Perp Figure 10.7: KT; distributions for data (histogram) and Pythia (points). This quan- tity is related to the detector 7) resolution which is effected by radiation. 204 figure 10.7 continued. JOWO 0.22 . 0.2 0.18 + 0.1 6 0.1 4 m 0.12 z 0.1 0.08 0.06 0.04 0.02 %jlllof1jlll Jet100 0.2‘ ' . u 0.5 KT Perp .4 0.25 0.2 8 0.1 5 3 0.1 0.05 .IIIIIIIIIIIIIIIIIIIII TTI+ .4 4 J lJlllllllllllllllllllJlllll o° 0.5 KT Perp 205 Jet20 0.0355— + —: 0.033— \. 1+, ll. —E I o“ '21 2 0.025:— . ¥ , i —: I c l 3,. 2 g 0.02:— ‘ .. —: z 5 . .+ + a o.o15_— fl,‘ + __ 0.013— +. —: 0.005%— } "a —: 00— 10.1.”21:21H‘IHHIHHIHHIHI.0:7HH()I8””0:9.“‘ emf Jet50 I 'l' 'l' I "I 'I"'*I""rfi"lr _ 0.03 r Qua, ‘ —: 0.025:— ‘ in —f 3 0.023— _: 2 0.0153— ; _: 0.013— —f : ‘. : 0.005:— , —: c: T‘IZIIIIIIIIIIIIII..1.IIIIIIIIIIIIIIIIIIIIII° (l 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 emf Figure 10.8: Fraction of the total energy deposited in the electromagnetic compartment of the calorimeter for data (histogram) and Pythia (points). 206 figure 10.8 continued. Jef70 0.03 . é a r»; 0.025 ‘ . § 0.02 2 0.015 0.01 0.005 . o.” IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII LLLLlllllllllllllllllllllllllll 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 emf 207 Jet20 1_" l"| l l l l’ l "= _1: : 10 E 1: _2: : 10 .r a _3: : 10 r 1. .4: : 210 r 1E .5: 2 10 a" E _6: : 1° 5 1. '7 :I.IIII.IIIIIIIIIIIIIIIIIIIIIIIII.I|..I.I: 10.4 .3 .2 -1 0 1 2 3 4 Y Jet50 1E""l""|""l‘"'l""l""lrj"l""= .1: 2 10 gr 1: .2: 2 1° 5 1. _3: : "10 r 25 I: E 10 g ‘g _5: : 10 I; 1: 45: E 10 g" 1: '7 ":IIIJI I .I III: 10.4 .3 -2 -1 0 1 2 3 4 Y Figure 10.9: Rapidity (Y) distribution for data (histogram) and Pythia (points). 208 figure 10.9 continued. Jet70 ..L O [1 III"! III"! III"! III"! III"! I I. lllllllllllllllll L Illll -< #:4de lLlllJ lelld Lilli Jllll‘ [1111‘ HI] . lll'l Illll IIIIIIII'IIIIIIII lllllll L_lllllq ”NH ”I"! Hm” lllllq Illlll‘ Ii ‘ 209 < ‘LIIIIIJ lllll‘ llllld llllll‘ llllld lllll‘ llll Jet20 [171—Fr llllllr I I £10" 2’ — -3 1° -1 I i Jet50 1 10 f 'r' fil'” I' ' l' 'l '1' I I: ‘3 -2 2310 E— E )— _I -3 10 .1 ‘7' ‘1’ Figure 10.10: Azimuthal angle ((15) distributions for data (histogram) and Pythia (points). 210 figure 10.10 continued. Jet70 10' 1 10‘ Jet100 1 0 1O fiIIIIIIIIIIIII'IITIlIIIIIIIIIIIIIIIIIIIL - ,— _ u— _ — —I I— —I I. —I _ —I I— _I I— _ lllllllllll‘llllllllllllllllllllIllllrlll -1 0 1 2 3 4 5 6 7 .I IIIIIIT—TTFIIIIIITIIITITrIl—l‘l—[IIIIlllll llllll lllllllllllllllllllllllllIllllllllllllll -1 0 1 2 3 4 5 6 7 ¢ 211 ITIIIIIIIITTTI[ITIIITTTIIflIIITIII 10 Mldpolnt cone R=0.7 , fm=0.5 1 I L = 218 pb" A g 10.1 0.1<|Y|<0.7 Q -2 p a 10 ythla abs. norm. 5 I l- 10 g 4 >- 10 - U : - \ .5 - " q: 10 E I :t 5% Sys. Pt Scale uncert. '5: 1 0'6 é— — Hadron level Pythia (CTEQ5L) E: 10'7 : 0 Data corrected to Hadron Level. % 10.81] lllllllllllilllllLlLlllllll [ll 0 100 200 300 400 500 600 700 PT (GeV) Figure 10.11: Data corrected to hadron level versus hadron level pythia (CTE'Q5L): log scale. The band represents the 21:5‘70 energy scale systematic uncertainty. Both the data and Pythia still have underlying event present. 212 (D IllIIlllllllllllllllllllllilllllT Midpoint cone R=0.7, f =0.5 merge I L = 218 pb“ 0.1<|Y|<0.7 Pythia abs. norm. 0 Data Corr. to Hadron Level/Hadron Level Pythia. I i 5% sys. Pt scale uncert. Data/Pythia(CTE05L) CD ""l"‘rl""l""l"" IIIIIIIIIIIIIIIIIIIIIIII — l l I l llllllllliIlllllllllllllllllllllll 0 100 200 300 400 500 600 700 PT (GeV) Figure 10.12: Data corrected to hadron level versus hadron level pythia (CTEQ5L) linear scale. The band represents the i5% energy scale systematic uncertainty. Both the data and Pythia still have underlying event present. 213 P1M5”— P144” o(hadron)/Pythia(CTEQ5l) Stat.Err 61-67 1.33421 0.0345388 67—74 1.26632 0.034378 7481 1.31026 0.0367131 81-89 1.23448 0.0409693 89-97 1.31345 0.0260076 97-106 1.32763 0.0172449 106-115 1.35039 0.0190693 115.125 1.34406 0.0192351 125.136 1.37589 0.0207802 136-158 1.41256 0.0149788 158-184 1.43318 0.015308 184212 1.45645 0.0177192 212-244 1.48374 0.019663 244280 1.49944 0.0247599 280-318 1.62059 0.0397356 318—360 1.62995 0.0643615 360—404 1.54883 0.109849 404464 1.71677 0.2092 464530 2.78737 0.677675 530—620 1.47729 1.47763 Table 10.1: Ratio of measured hadron level inclusive jet cross section over hadron level Pythia cross section. This cross section still contains underlying event. 214 Chapter 11 Comparison of the Data to NLO pQCD 11 .1 Introduction The corrected inclusive jet cross section is compared to NLO pQCD predictions from the EKS program [29, 30, 31] using the CTEQ6.1 PDF set. In all comparisons that follow, the EKS prediction uses a renormalisation scale and factorisation scale of leet/Z. The CTEQ6.1 PDF’s are an update to the published CTEQ6M PDF sets. The CTEQ6.1 PDF set contain the CDF and DO Run 1 jet data and are thus the most complete/ up to date set. The predictions for NLO pQCD depend on input parameters such as the parton distribution functions, choice of factorisation/renormalisation scales and the choice of as(Mz). As discussed previously it is desirable to have the clustering at parton level, hadron level and calorimeter level done in a consistent way. The clustering at the calorimeter level is performed over jets of hadrons (seen as calorimeter towers). The edges of these jets are not distinct. Some events will have jets that are close to one another, introducing ambiguities such as those seen with merging and splitting in jet algo- rithms. These ambiguities are not modelled in the NLC) pQCD predictions as there 215 are, at most, 3 partons in the final state. The parameter Rsep was introduced into the theoretical prediction to approximate the effects of merging and splitting of clusters as done by experimental algorithms. The choice of ,u scale introduces an unavoidable uncertainty in a fixed order per- turbative calculation. Traditionally, u = PT,“ / 2 has been used but other scale choices such as u = PT,“ are also acceptable. 11.2 Correcting NLO pQCD for Underlying Event In order to compare N LO pQCD to the hadron level data the underlying event must be accounted for. This can be done either by subtracting the UE contribution from the data or adding it to the NLO predictions. Here the later method was used. The underlying event corrections were derived from both the Pythia and Herwig in the form of bin by bin corrections much like those used for the smearing correction. The corrections derived from Pythia were used to correct the data as these provided a consistent scheme with respect to the calorimeter to hadron level corrections. There are two competing effects that make up the parton to hadron level correc- tion: the first is fragmentation and the second is underlying event. Fragmentation causes energy to be lost from the jet cone whereas the underlying event adds ad- ditional energy into the jet cone. In Pythia the underlying event is the dominant process, especially at low PT. In Herwig the underlying event contribution also domi- nates but not as much. The difference in the overall parton to hadron level correction is associated with two effects: Herwig underlying event is smaller than Pythia and Pythia has a more physical implementation of the underlying event (it is included at parton level). 216 11.3 Results In this section we present a comparison of the data corrected to the hadron level and the parton level to the NLO pQCD prediction from the EKS program. The comparison of the hadron level cross section to the NLO pQCD prediction is done in three steps: the first is the comparison with no underlying event correction, the second is the comparison with the underlying event correction and the third is the comparison with the full parton to hadron correction. The results are organised in the following way: 0 Figures 11.1 and 11.2 show the data (corrected to hadron level) compared to the NLO pQCD prediction. No correction has been made for the underlying event. 0 Figures 11.3 and 11.4 show the data compared to the NLO pQCD prediction. The prediction has been corrected on a bin by bin basis to account for the underlying event only. 0 Figure 11.5 shows data compared to the NLO pQCD prediction with the un- derlying event and fragmentation correction applied. 217 llllllllTrTllllTTllllIlllllllllll Theory: NLO poco EKS one 6.1, (user/2), nw=1 .3 10 Data: Midpoint cone R=0.7, fm=0.5 1 Data corrected to hadron level A 3 1o" 0.1<|Y|<0.7 c; I 3 10-2 IL=218 pb 5 .3 )- n.1--10 E- ; 10" P .0 \ -5 Nb 10 U .5 'Total Sys Unoert. 10 - Data corrected to hadron level 10'7 — NLO pQCD, EKS(CTEQG.1) 10.8LL llllllllllllllllllllJlLLllllll 0 100 200 300 400 500 600 700 PT (Gev) Figure 11.1: Data corrected to hadron level versus NLO pQCD. The data is corrected for multiple interactions; however no correction for underlying event has been made. 218 3'5 IllIITITTTIITIITIIIIIIIIIIIIIIIl Theory: NLO poco EKS creo 6.1, (”pf/2), R9513 Data: Midpoint cone n=0.7, fmm=0.5 Data corrected to hadron level. 0.1<|Y|<0.7 l9 or I L = 218 pb“ N —l Ul lllllllllllllllllllllll llllllllllllllllllllllLl 6 Ratio (Data/CTE06.1) 1 _ .- 0'5} .Total Sys Uncert. on corrected Data '_' 0:— . Data/NLO pQCD j _ I I I I l I I I I l I l I I l I I I I l I I I I I I I I I l L I— 0 1 00 200 300 400 500 600 PT (GeV) Figure 11.2: Data corrected to hadron level versus NLO pQCD. (no underlying event correction) 219 3'5 I I I I I I I I I I I I I I I I I Ij I I I I I T _ I | I I T I _ I Theory: NLO poco EKS cnao 6.1, (u=P3.’-“I2), 35.943 I 3 — Data: Midpoint cone n=0.7, fmm=0.5 —_ A I Data corrected to hadron level + UE event corr. : F _ _I “5 25 _ 0.1<|Y|<0.7 fl 3 I I L = 218 pb" I I- 2 '_ _‘ % z i '5 15: 2 D ' - - V - _ _O - _ 5 1 (I : I I _ _ b 0.5”— —‘ : .Total Sys Uncert. on corrected Data 2 0 — o Data/NLO pQCD - -J_ I I I I I I I I I l I I I I I I l I I I I I I I I I I l I l I- o 100 200 300 400 500 600 PT (GeV) Figure 11.3: Data corrected to hadron level and corrected for underlying event versus NLO pQCD. 220 3: IVIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII Theory: NLO poco EKS CTEG 6.1, (reef/2), nw=1.3 Data: Midpoint cone R=0.7, tm=0.5 Data corrected to hadron level 0.1<|Y|<0.7 N or I L = 218 pb" N —l 0| IIIIIIIIIIIIIIIIIIIIIII IIIIIIIIIIIIIIIIIIIIIIII 0 Ratio (Data/CTE06.1) 1 ._ _I 0'5 _— .Total Sys Uncert. on corrected Data 1 I . Data/NLO pQCD Before UE Corr. 3 O l . Data/NLO pQCD After UE Corr. _— _ I I l I I I I I I I l I l l I I I l [J I I I I I I I I I I I I— 0 1 00 200 300 400 500 600 PT (GeV) Figure 11.4: Data corrected to hadron level before and after underlying event correc- tion versus NLO pQCD. 221 0 Ratio (Data/CTE06.1) 3!: IV IIIIIIIIIIIIIIIIIIIIIIIIIIIIII r C Theory: NLO poco EKS CTEQ 6.1. (u=P$"/2), 3351.3 I 3 4 Data: Midpoint cone n=0.7, fmm=0.5 —_ C Date corrected to parton level : 2_5 '_ 0.1<|Y|<0.7 i I I L = 218 pb" : 2 '_ .1 1.5; —‘ 1 _ _ 0'5 _— .Total Sys Uncert. on corrected Data 1 I j 0 —_ . Data/NLO pQCD _— - I I I I I I I I I I l l I I I I I I l I I I I I I I I I I I l l- 0 100 200 300 400 500 600 PT (GeV) Figure 11.5: Data corrected to parton level versus NLO pQCD. 222 PT Bin Had. Level Data Stat. Err. UE Corr. Part. Corr. NLO pQCD 61-67 11.512 0.3040 1.2733 1.22425 8.03309 67-74 6.30843 0.1692 1.2404 1.19712 4.6274 74-81 3.5305 0.1026 1.2112 1.17312 2.6693 81-89 1.9922 0.0638 1.1857 1.15202 1.5570 89—97 1.1307 0.02194 1.1631 1.13341 0.9123 97-106 0.6588 0.008480 1.1434 1.11709 0.5407 106-115 0.3869 0.005502 1.1261 1.10272 0.3222 115-125 0.2283 0.003243 1.1109 1.09013 0.1940 125-136 0.1345 0.00204 1.0970 1.07857 0.1148 136-158 0.06329 0.0006792 1.0800 1.06435 0.05528 158-184 0.02273 0.0002429 1.0620 1.04918 0.02038 184-212 0.007999 9.7479e-05 1.0482 1.03747 0.007330 212-244 0.002853 3.728e-05 1.03831 1.02882 0.002604 244-280 0.0009352 1.5562e-05 1.03138 1.02259 0.0008745 280-318 0.0003234 7.9200e-06 1.02734 1.01869 0.0002834 318-360 0.0001014 4.0050e-06 1.02575 1.01674 8.8620e-05 360-404 2.8252e—05 2.0023e-06 1.02622 1.01644 2.6066e-05 404-464 7 .3605e—06 8.9697e-07 1.02892 1.01782 6.3116e-06 464-530 1.9024e—06 4.6254e—07 1.03424 1.02118 1.0685e-06 530-620 9.9217e-08 9.9239e-08 1.04297 1.02709 1.1929e—07 Table 11.1: The inclusive jet cross section corrected to the hadron level and NLO pQCD prediction. The additional columns are the underlying event correction factors (UE. Carr) and the parton to hadron correction factors (Part. Corr). 223 PT Low - PT High 051%: 0" /o£VQLgI)D Stat. Err +Sys(-Sys) 61-67 1.365 0.036 +0.284(-0.243) 67-74 1.298 0.035 +0.286(-0.241) 74-81 1.260 0.037 +0.290(-0.241) 81-89 1.219 0.039 +0.294(-0.241) 89-97 1.180 0.023 +0.300(-0.243) 97-106 1.160 0.015 +0.306(-0.245) 106115 1.144 0.016 +0.313(-0.248) 115-125 1.120 0.016 +0.322(-0.252) 125-136 1.115 0.017 +0.335(-0.256) 136-158 1.090 0.012 +0.354(-0.262) 158-184 1.062 0.011 +0.377(-0.273) 184-212 1.039 0.013 +0.402(-0.285) 212-244 1.043 0.014 +0.431(—0.298) 244-280 1.018 0.017 +0.462(-0.314) 280—318 1.087 0.027 +0.496(—0.331) 318—360 1.090 0.043 +0.533(-0.350) 360—404 1.032 0.073 +0.578(-0.370) 404-464 1.111 0.135 +0.632(-0.395) 464-530 1.696 0.412 +0.699(-0.425) 530-620 0.792 0.792 +0.701(-0.462) Table 11.2: Data corrected to the hadron level over NLO pQCD including full statis- tical and systematic errors (No underlying event correction has been applied). 224 PT Low - PT High oggm‘iUE/oég’cop Stat. Err +Sys(-Sys) 61-67 0995' 0.044 +0.438(-0.413) 67-74 1.011 0.047 +0.409(-0.381) 7481 1.014 0.050 +0.386(-0.354) 81-89 0.977 0.050 +0.369(-0.333) 89-97 1.013 0.037 +0.357(-0.316) 97-106 1.006 0.028 +0.349(-0.303) 106115 0.995 0.031 +0.344(-0.294) 115-125 0.982 0.031 +0.343(-0.287) 125-136 0.993 0.032 +0.344(-0.283) 136158 0.979 0.024 +0.350(-0.281) 158-184 0.966 0.022 +0.363(-0.284) 184212 0.976 0.025 +0.382(-0.291) 212—244 0.998 0.022 +0.405(-0.303) 244280 0.966 0.026 +0.433(-o.317) 280—318 1.062 0.034 +0.463(-0.333) 318-360 1.059 0.046 +0.497(-0.351) 360—404 0.992 0.074 +0.534(-0.371) 404464 1.072 0.133 +0.579(-0.396) 464530 1.639 0.401 +0.633(-0.426) 530-620 0.757 0.758 +0.70l(-0.465) Table 11.3: Data corrected to the hadron level over NLO pQCD including full statis- tical and systematic errors. The ratio has been corrected for underlying event. 225 PT Low - PT High ogggm/opl‘g’gp Stat. Err +Sys(-Sys) 61-67 1.117 0.042 +0.438(—0.413) 67-74 1.099 0.044 +0.409(-0.381) 7481 1.115 0.050 +0.386(-0.354) 81-89 1.054 0.044 +0.369(—0.333) 89-97 1.088 0.030 +0.357(-0.316) 97—106 1.085 0.027 +0.349(-0.303) 106115 1.069 0.030 +0.344(-0.294) 115-125 1.050 0.030 +0.343(-O.287) 125-136 1.054 0.031 +0.344(-0.283) 136158 1.036 0.022 +0.350(—0.281) 158184 1.016 0.021 +0.363(-0.284) 184212 1.023 0.026 +0.382(—0.291) 212-244 1.036 0.023 +0.405(-0.3o3) 244-280 1.011 0.027 +0.433(-0.317) 280—31 8 1 .096 0.034 +0.463 (-0.333) 318360 1.098 0.049 +0.497(-o.351) 360-404 1.025 0.076 +0.534(-0.371) 404-464 1.108 0.137 +0.579(-0.396) 464-530 1.683 0.411 +0.633(-0.426) 530-620 0.781 0.781 +0.701(-0.465) Table 11.4: Data corrected to the parton level over NLO pQCD including full statistical and systematic errors. 226 11.4 Conclusion In summary, we have measured the inclusive jet cross section in the PT range 61-620 GeV. The statistical uncertainty of the data is significantly better than the system- atic uncertainty in the measurement. We see good agreement between the central values of data and NLO pQCD predictions. The systematic uncertainties quoted here very conservative. From the figures 11.1- 11.5 it is seen that within the system- atic uncertainties the corrected data agree very well with the NLO pQCD prediction (CTEQ6.1). The N LO pQCD predictions use the CTEQ6.1 PDF set. These contain jet data both CDF and D0 run I jet data. 227 Chapter 12 Systematic Uncertainties 12.1 Introduction The systematic uncertainties on the measured inclusive jet cross section come from three sources: calorimeter response, resolution and luminosity. The uncertainty on the luminosity has no PT dependence: it only effects normalisation. Systematic uncertainties arising from the unfolding of the measured cross section to the hadron level contributes to the uncertainty on the corrected inclusive cross section. This contribution is in addition to the systematic uncertainty associated with the measured cross section. There are additional uncertainties on the NLO pQCD due to choice of [1 scale, PDF uncertainty and the underlying event correction. 12.2 Jet PT Scale Uncertainties The uncertainties associated with jet energy scale include calorimeter response, frag- mentation tuning, multiple interaction energy and the energy from the underlying event falling into the jet cone. 228 12.2.1 Calorimeter Response The uncertainty on the calorimeter response is taken from a comparison of data and Pythia. The comparison is made between the quantity R(PT) = 2 PT(Cal)/ Z PT(Tracks) (12.1) for data and Pythia. The double ratio R(PT)Data/R(PT)Pythia is then found. The deviation of this double ratio from unity is used to set the jet scale uncertainty. From this study the uncertainty is found to be 5% [35]. The non linearity of the central calorimeter to charged particles contributes to the energy loss of the calorimeter. Jet fragmentation is a measure of the distributions of the charged particles associated with tracks and therefore effects how well the simulation reproduces the jet energy and jet energy losses in the detector. The overall agreement between the data and Pythia is reasonably good (seen in jet shapes). The uncertainty on the cross section due to the fragmentation tuning is related to the track finding efficiency in the dense track environment of jets. The differences between track finding efficiencies in data and Pythia are not corrected for in the determination aformentioned 5% systematic, therefore this value is a conservative estimate of the systematic uncertainty. 12.2.2 Unfolding In order to estimate the uncertainty associated with the hadronisation model used in the Monte Carlo corrections we treat the difference between the bin correction factors from Pythia and Herwig as a systemaic uncertainty. The difference is only significant below 100 GeV. 229 12.2.3 Multiple Interaction Uncertainties The contribution to the jet energy due to multiple interactions is removed from the raw jets. For a given event if more than one quality 12 vertex is found 0.92 GeV is removed for each additional vertex. The systematic uncertainty on this correction is 30%: this systematic covers the luminosity dependence of the measurement and also the changes seen by increasing/ decreasing the tower threshold by 50 MeV in the measurement definition. 12.2.4 Underlying Event Underlying event corrections are derived from Pythia and Herwig. We used the cor- rection derived from Pythia to correct the data as this provides a consistent correction scheme for calorimeter to hadron corrections and the parton to hadron corrections- Also Pythia includes the underlying event contribution at parton level which Herwig does not. This is important when deriving parton-hadron level corrections. The dif— ference between Herwig and Pythia corrections are used as the systematic uncertainty. It is not combined with the uncertainties on the measured cross section. Instead it is treated as a systematic on the NLO pQCD prediction. 12.2.5 Integrated Luminosity and Z Vertex Uncertainty The integrated luminosity enters the cross section expression as a normalisation fac- tor. The uncertainty on the integrated luminosity contributes to the systematic un- certainty on the cross section as follows: do 6L The uncertainty on the luminosity is 6.0% common for all PT values. 230 12.2 .6 Results Y I I If T Ifi r j r I I I I I V I I V I I Y fif Y I rfifi 0 I I I W I I UEMSyaErr . . P. 3 4.050IIII160IIII260IIII3®IIIImIIIIséoILII6®J4II7m P,(Gev) 0.2 V V V V fr V V V I V V V V l’ T Y V ‘i T V f T 1 ‘l V V V V I V W V v 0.1 5 0.1 0.05 -0.05 -0.1 -0.15 Resolution Sys. Err. o lilillllllIlillllilllliilllllllllllllIi .3 '8‘: g- §3 g. §. §: I 0.2 I I V 'TT ‘fij r' I I 7T1 I I 71". rrf' U I I I l I (L15 on (L05 .0 4L05 -0J 4L15 -0.2t Luminosity Sys. Err. II A L l I l A l J I I A l A I l I I I I l A L4 I 100 200 ‘ "300" "400 500 l ' IIIIIIIIIIIIIIIIIII'IIIIIIIII[IIIIIIIII Figure 12.1: Fractional contributions to the total systematic uncertainty on the inclu- sive jet cross section. We see that the energy scale and underlying event uncertainties dominate the total systematic uncertainty. 231 figure 12.1 continued. 0.2 0.1 5 0.1 0.05 ..o -0.05 -0.1 -O.15 41.2" Unfolding Sys. Err. I rillIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII d 1 EV §. § 0.8 0.6 0.4 0.2 oz -o.4 -o.s -o.s UE Sys. Err G ' hllllllllllillillllllillllilIllllIilllll 0.8 0.6 0.4 0.2 -o.2 -o.4 -o.e -o.s E-oooio a. Total Sys. Err. O Illlllllllllillllllllli'lllllillllill" I :5" L 700 Pt (Gav) 232 PT Low - PT High E—scale(+) E—scale(-) 61-67 0.255857 -0.214444 67-74 0.261371 -0.217418 74-81 0.267309 -0.220621 81-89 0.27367 -0.224053 89-97 0.280456 -0.227713 97-106 0.287666 -0.231603 106-115 0.2953 -0.235721 115-1 25 0.303359 -0.240067 125-136 0.312265 -0.244872 136-158 0.326261 -0.252422 158-184 0.346619 -0.263403 184-212 0.369521 -0.275757 212-244 0.394968 -0.289484 244-280 0.423809 -0.305041 280—318 0.455193 -0.321971 318-360 0.489123 -0.340274 360-404 0.525597 -0.359949 404-464 0.569706 -0.383742 464-530 0.623145 -0.412568 530—620 0.689307 —0.448258 Table 12.1: Energy scale systematic uncertainty. 233 PT Low - PT High Unfolding(+) Unfolding(—) 61-67 67-74 74-81 81-89 89—97 97-106 106-1 15 115-125 125-136 136—158 158-184 184-212 212-244 244-280 280-318 318-360 360-404 404-464 464-530 530-620 0.0840673 0.0737525 0.064686 0.056741 0.0497957 0.0437369 0.0384625 0.0338817 0.0297355 0.0246824 0.019571 0.0159109 0.0135178 0.0121957 0.011868 0.0123901 0.0136477 0.0158327 0.0191441 0.023914 -0.082172 -0.0708698 -0.0609694 -0.0523296 -0.0448149 -0.0382995 -0.0326698 -0.027 8248 -0.0234897 -0.0183037 -0.0132506 -0.00989093 -0.00801959 -0.00744484 -0.00805544 -0.00969023 -0.0122223 -0.0160229 -0.02l3666 -0.0287277 Table 12.2: Hadronisation model systematic uncertainty. 234 PT Low - PT High UEM(+) UEM(-) 61-67 0.0132962 0.0112895 67—74 0.0121768 0010397 74-81 0.0111847 000960322 81-89 0.0103068 000889782 89-97 0.00953027 000827081 97—106 0.00884327 00077129 106-115 0.00823514 000721573 115-125 000769638 000677185 125-136 000719679 000635643 136-158 0.0065646 000582353 158-184 000587912 000523199 184-212 000532653 000473791 212-244 000488753 000432619 244-280 000453533 000397355 280-318 000426825 000368092 318—360 000407113 000343642 360-404 000393205 000323069 404-464 0.0038333 000303637 464-530 000378311 000285541 530-620 000379103 000268631 Table 12.3: Multiple interaction systematic uncertainty. 235 PT Low-PT High UE(+) UE(-) 61-67 0.334386 0.334386 67-74 0.294689 -O.294689 74—81 0.25965 025965 81-89 0.228791 0.228791 89-97 0.20165 0.20165 97—106 0.177801 0.177801 106-115 0.156859 0156859 115125 0.138479 0.138479 125-136 0.121627 0.121627 136-158 0.100669 0.100669 158-184 0.0786399 0.0786399 184-212 0.0617522 00617522 212244 0.0493094 0.0493094 244-280 0.0404576 0.0404576 280—318 0.0350219 0.0350219 318-360 0.0324551 0.0324551 360-404 0.0323262 0.0323262 404-464 0.0346786 -0.0346786 464—530 0.0400378 0.0400378 530—620 0.0492035 0.0492035 Table 12.4: Underlying Event systematic. 236 12.3 Jet Energy Resolution Uncertainty The uncertainty on the jet resolution is determined from the average deviation of the single jet resolution (GEMS) measured in the data from that measured in Pythia. The comparison is made over 12 bins of (PT). This uncertainty is propagated into the cross section using the following procedure: the systematic uncertainty for the resolution is found by smearing the true hadron level inclusive jet PT distribution by: Cal P N¥74(P¥“d)’= Nit. “(P PH““')(1 +a( P1304 ) x a), (12.3) where N Jet (P H ad) and Nfigd(P1gad)’ are the number of hadron jets in a bin before Cal and after smearing respectively. 4%) is the jet resolution as a function of P11!“ T and a is a random number generated from a gaussian distributions with u = 0 and o = 1,108 and 0.92: these are labelled gaus(0,1), gaus(O, 1.08) and gaus(0,0.92) respectively. The size of the variation of the 0’s comes the average Pythia-Data oJet discrepancy. Having smeared the inclusive PT distribution the overall change in cross section was determined from the ratio: 0,1 NJet/N N32?“ ) Ngaus(,(0, 1.92) Ngaus(0,1) 12 4 Ngaus(0,l.08) Jet / ( - ) NJBt/N Jet where N Jet is the number of jets in a PT bin without smearing, Nggi‘sm’ 1) is the number of jets in a PT bin after the resolution has been smeared with a gaussian (,u = 0 and a = 1) and N33 3(0’1'08) is the number of jets in a PT bin after the resolution has been smeared with a gaussian (u = 0 and o = 1.08). This procedure is repeated with a (u = O and o = 0.92) gaussian. These two ratio’s give the fractional 237 uncertainty on the cross section due to the resolution difference in the data and Pythia. 238 q .1 IIIIIIIIIIIIjTITIIIII'IIIIIIIIrTIIIII 1 .4 1.2 0 Fractional Difference in arms: data and Pythia Fit to P0 1 p0 005714 1 000363 F a P N lllllijllllJLLllllllllllllllllllllllll (Om-Wyonfl P P «D Q [IllllrIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII $85 llllllllllJllLLlJJllllqulllll LL11 ‘50 100150 200 250 300 350 400 (P'rHGeVl p Figure 12.2: Data-Pythia Resolution difference versus (P11? ij ct) 0.2 I I I I I I I I I I I I I I r T I I I I I I III I I I I I I 0.18 0.16 0.14 0.12 I A IIJUJILL on.) 9999 oSESSS G ..lLl [_IAJWIILlll TIIIIIIIIIIIIIIIIIIIIIWIIIIIIIIIII III 1 1l4lll¢Jllllllllilllllllllllll-l 1 00 200 300 400 500 600 P?“ (Gav) Figure 12.3: 0(fb) versus P11! adm". The errors are errors from the gaussian fit. This function is smeared with a gaussian to set the resolution uncertainty. 239 PT Low - PT High Resolution(+) Resolution (-) 61-67 0.0585414 -0.0494923 67-74 0.0535827 -0.045816 74-81 0.0494036 -0.0427523 81-89 0.0459318 -0.0402457 89-97 0.0430981 -0.0382426 97- 106 0.0408382 -0.0366936 106-11 5 0.039094 -0.0355538 115-125 0.0378138 -0.0347834 1 25— 136 0.0369202 -0.034334 136-1 58 0.0363474 -0.034269 158-184 0.036787 -0.0351 564 184-212 0.0384701 -0.0370714 212-244 0.0412917 ~0.0399332 244-280 0.04528 -0.0437862 280—318 0.0502519 -0.0484666 318-360 0.0561252 -0.0539107 360—404 0.0628349 -0.0600686 404-464 0.0713266 -0.0678065 464—530 0.0819923 -0.0774728 530—620 0.0955784 -0.0897343 Table 12.5: Resolution systematic uncertainty. 240 PT Low - PT High Total(+) Total(-) 61-67 0.437662 0.413194 67—74 0.40892 0.380713 74-81 0.386293 0.354019 81-89 0.369156 0.332539 89—97 0.356856 0.315683 97-106 0.348746 0.302863 100115 0.344213 0.293517 115-125 0.342695 0.287121 125-136 0.343805 0.283066 136-158 0.349505. 0.281059 158184 0.362905 0.283909 184-212 0.381734 0.291462 212-244 0.404899 0.302506 244-280 0.432516 0.316663 280318 0.46337 0.333046 318360 0.497207 0.351357 360404 0.533898 0.371451 404-464 0.57855 0.396137 464—530 0.632942 0.426477 530620 0.700633 0.464587 Table 12.6: Total systematic uncertainty on the inclusive jet cross section. 241 12.4 Sensitivity to Input PDF In order to check the sensitivity of the jet correction method to the input PDF we would have liked to have had multiple Pythia samples generated with different PDF’s. Generating a second complete set of Pythia samples was not practical so instead the standard Pythia CTEQ51 sample was re—weighted by the ratio of the LO pQCD cross section predictions from EKS using the CTEQSI and CTEle PDF sets. The re- weighting is performed on the PAT distribution. This re-weighting gives a new Pythia sample with a smaller high PT cross section and a larger low PT cross section with respect to Pythia CTEQSI. We refer to the new/re-weighted Pythia sample as Pythia CTEle. Having re-weighted the distribution, the calorimeter and hadron cross sections were found using the CTEle sample. The CTEle calorimeter cross section was corrected using the standard CTEQ51 jet corrections and the corrected cross section compared to the expected/true CTEle hadron cross section. Figure 12.8 shows that within the statistical uncertainty the correct hadron distribution is recovered. 242 d IIIIIIIIllIIII'IIIIIIIrIlIIIIlIIIIIIIII “Imam ddd-fl-fi‘dd OOOOOOOO behé-Lbh-h '1‘ ..Q 1* 10 ~ ‘l lllllllllllllllllLlllllllllLlllllJ”l 0 1 00 200 300 400 500 600 700 800 ward (G ev) Figure 12.4: P711 Pythia CTEQ5I Monte Carlo and Pythia re-weighted to resemble Pythia C'TEle. 243 q IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIrTIII 1 1.4 1.2 0.8 0.6 0.4 NEvenu(ML)INEvente(5 L) 0.2 TIIIIIIIIIILIIIIIIIIIITIIIIF lllllllllllllllllllLlllllllLl o lllllllllllLlllllllllLLJlLllllllll 100 200 300 400 500 600 700 800 P}(GeV) 0 Figure 12.5: Ratio of RT Pythia CTEQ5I Monte Carlo and Pythia re-weighted to re- semble Pythia CTEle. It is clear that CTEQ5I has a much stifier gluon distribution giving a larger high PT cross section. 244 IIIIIII'TIIIIIITITTITTIIIII'IIII 13 10 a 12 10 11 :3... — Pythia CTEQSL ----- Reweighted I‘.. .'~ ‘~ . lllllll llllLLllll ll l l l I I lJLllll l ‘11‘ 0 100 200 300 400 500 600 700 PT (GeV) Nah/(2 GeV) d d .5 .5 d d d d d -| O Oucac‘OuOGONOOOo Figure 12.6: Calorimeter level inclusive jet distribution for Pythia CTEQ5l and Pythia re-weighted by CTEle 245 1o - IITIIfiIrrIITIIIIIIIIIIII'IIIIIIII 10 — Pythia CTEQSL ----- Reweighted Mon/(2 GeV) N a b on O N O O | | ‘0 ~ 0 ddd—l-L-A-ld-l OOOOOOOOO 'I. 1 lllllLllllJJLllLLlllJl111L4411141 0 100 200 300 400 500 600 700 P?“ (GeV) Figure 12.7: Hadron level inclusive jet distribution for Pythia CTEQ5I and Pythia re-weighted by CTEle 246 0.1 I I I I I I I I I I I I I I I I I I I I I I I I I I I r I I I 0.08 0.06 0.04 0.02 I _ll ll -0.02 Ratio of CTEQSIJCTEOML Unfolding .5 .6 55 8 8 E o IITITIIIIIIIIIIIIITI‘IIIIIIIIIIIIIII 1 III I'llllllLIlllllllIlll lJlIlllllllIlllllLl' I I L I I I I I I I I I l I I I I I I I I l I I l I I I I I I 1 00 200 300 400 500 600 PT (GeV) :5 c—l Figure 12.8: The ratio of the Pythia re-weighted by CTEle corrected to hadron level using Pythia CTEQ5l jet corrections over the true hadron level cross section. Within statistical uncertainties the true cross section is recovered. 247 Bibliography [1] C. N. Yang and R. L. Mills, Phys. Rev. 96 191 (1954) [2] R.K. Ellis, W.J. Sterling and RR. Webber ” QCD and Collider Physics” Cam- bridge Monographs on Particle Physics, Nuclear Physics and Cosmology, Cam- bridge Univ. Press, United Kingdom (1996). pp 8,9. [3] R.K. Ellis, W.J. Sterling and RR. Webber ” QCD and Collider Physics” Cam- bridge Monographs on Particle Physics, Nuclear Physics and Cosmology, Cam- bridge Univ. Press, United Kingdom (1996). pp 260. [4] R.K. Ellis, W.J. Sterling and RR. Webber ” QCD and Collider Physics” Cam- bridge Monographs on Particle Physics, Nuclear Physics and Cosmology, Cam- bridge Univ. Press, United Kingdom (1996). pp 17. [5] R.K. Ellis, W.J. Sterling and RR. Webber ” QCD and Collider Physics” Cam- bridge Monographs on Particle Physics, Nuclear Physics and Cosmology, Cam- bridge Univ. Press, United Kingdom (1996). pp 27. [6] R.K. Ellis, W.J. Sterling and RR. Webber ” QCD and Collider Physics” Cam- bridge Monographs on Particle Physics, Nuclear Physics and Cosmology, Cam- bridge Univ. Press, United Kingdom (1996). pp 18. [7] R.K. Ellis, W.J. Sterling and RR. Webber ” QCD and Collider Physics” Cam- bridge Monographs on Particle Physics, Nuclear Physics and Cosmology, Cam- bridge Univ. Press, United Kingdom (1996). pp 248,249. [8] R.K. Ellis, W.J. Stirling, B.R. Webber. http: / / www.hep.phy.cam.ac.uk/ theory/ webber/ QC [9] L.D. Faddeev and V.N. Popov, Phys. Lett. 325, 29 (1967). [10] G. ’t Hooft and M. Veltman, Nucl. Phys. 50, 318 (1972). [11] G. Leibbrandt, Rev. Mod. Phys. 59, 1067 (1987). [12] BL. Combridge, J. Kripfganz and J. Ranft, Phys. Lett. B70 (1977) 234. [13] MG. Albrow et al., Nucl, Phys. B160 (1979) 1. [14] A.L.S. Angelis et al., Phys. Scr. 19 (1979) 116. [15] A.G.C1ark et al., Nucl. Phys. 160 (1979) 397. [16] D. Drijard et al., Nucl, Phys. B166 (1980) 233. 248 [17] G. Sterman and S. Weinberg, Phys. Rev. Lett. 39, 1436 (1977). [18] Yu.L. Dokshitzer, D.I. Dyakanov and 8.1. 'fioyan, Phys. Rep. 58C, 269 (1980). [19] P.Bagnaia et a1, Measurement of Jet Production PrOperties at the CERN pp Collider, Phys. Lett. 144B 283 (1984). [20] M.Banner et al., (UA2 Collaboration), Phys. Lett. 118B(1982) 203. [21] G. Arnison et al., (UAl Collaboration), Phys. Lett. 123B(1983) 115. [22] R.K. Ellis, W.J. Sterling and RR. Webber ” QCD and Collider Physics” Cam- bridge Monographs on Particle Physics, Nuclear Physics and Cosmology, Cam- bridge Univ. Press, United Kingdom (1996). [23] S.Ellis, Z. Kunszt and D. Soper, Phys. Rev. Lett. 62 2188 (1989); Phys. Rev. Lett. 64 2121 (1990); Phys. Rev. D. 40 2188 (1989) [24] BL. Combridge, J. Kripfganz and J. Rantf, Phys. Lett. 70B(1977) 234. [25] F. Aversa, P. Chiappetta, M. Greco, P. Guillet, Phys. Lett. 1233 225 (1988);211B, 465 (1988); Nucl. Phys. B327, 107 (1995). [26] W.K. Sakumoto and A. Hocker, “Event |tha:| < 60cm Cut Efficiency for Run II”, CDF/ANAL/ELECTROWEAK/CDFR/6331. [27] M.Banner et al., (UA2 Collaboration), Phys. Lett. 118B(1982) 203. [28] G.Marchesini et al., Comp.Phys. Commun. 67, 465 (1992). [29] S.Ellis, Z. Kunszt and D.Soper, Phys. Rev. Lett.62 2188 (1989). [30] S.Ellis, Z. Kunszt and D.Soper, Phys. Rev. Lett.64 2121 (1990). [31] S.Ellis, Z. Kunszt and D.Soper, Phys. Rev. D40 2188 (1989). [32] J .F. Arguin, B. Heinemann, A. Yagil. Z-vertex efficiency, CDF—Note 6238. [33] Underlying Event Energy from Minimum bias Data. CDF / ANAL / J ET / CDFR/ 5008 Anwar Ahmad Bhatti [34] The CDF Collaboration, The CDF II Detector Technical Design Report. FERMILAB-Pub—96/390-E CDF [35] “Study of Jet Shapes in Inclusive Jet Production at CDF I ”, CDF/PHYS/JET/PUBLIC/6952 Mario Martinez. [36] “First Measurement of the Inclusive Jet Cross Section Using the Midpoint Al- gorithm”, CDF/ANAL/JET/ CDFR/ 7206 F. Chlebana, G. Flanagan, J.Huston, G. Latino 249 I[lllllllllll[ill]